text: Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscn2 Structured version   Visualization version   GIF version
  
  Theorem iscn2 20982
   Description: The predicate "𝐹 is a continuous function from topology 𝐽 to topology 𝐾." Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)
  Hypotheses
  Ref Expression
  iscn.1 𝑋 = 𝐽
  iscn.2 𝑌 = 𝐾
  Assertion
  Ref Expression
  iscn2 (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
  Distinct variable groups:   𝑦,𝐽   𝑦,𝐾   𝑦,𝑋   𝑦,𝐹   𝑦,𝑌
  
  Proof of Theorem iscn2
  Dummy variables 𝑓 𝑗 𝑘 are mutually distinct and distinct from all other variables.
  StepHypRef Expression
  1 df-cn 20971 . . 3 Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})
  21elmpt2cl 6841 . 2 (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
  3 iscn.1 . . . 4 𝑋 = 𝐽
  43toptopon 20662 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
  5 iscn.2 . . . 4 𝑌 = 𝐾
  65toptopon 20662 . . 3 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
  7 iscn 20979 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
  84, 6, 7syl2anb 496 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
  92, 8biadan2 673 1 (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
   Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2908  {crab 2912  ∪ cuni 4409  ◡ccnv 5083   “ cima 5087  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615   ↑𝑚 cmap 7817  Topctop 20638  TopOnctopon 20655   Cn ccn 20968 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-top 20639  df-topon 20656  df-cn 20971 This theorem is referenced by:  cntop1  20984  cntop2  20985  cnf  20990  cnima  21009  cnco  21010  ptpjcn  21354
   Copyright terms: Public domain W3C validator
explanation: The provided text is a formal proof in the Metamath Proof Explorer system. It defines and proves a theorem about continuous functions in topology. The theorem states the conditions under which a function \( F \) is considered continuous between two topological spaces \( J \) and \( K \). The text does not contain a question or an answer in the conventional sense; it is a structured proof with hypotheses, assertions, and steps.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscn2 Structured version   Visualization version   GIF version
  
  Theorem iscn2 20982
   Description: The predicate "𝐹 is a continuous function from topology 𝐽 to topology 𝐾." Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)
  Hypotheses
  Ref Expression
  iscn.1 𝑋 = 𝐽
  iscn.2 𝑌 = 𝐾
  Assertion
  Ref Expression
  iscn2 (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
  Distinct variable groups:   𝑦,𝐽   𝑦,𝐾   𝑦,𝑋   𝑦,𝐹   𝑦,𝑌
  
  Proof of Theorem iscn2
  Dummy variables 𝑓 𝑗 𝑘 are mutually distinct and distinct from all other variables.
  StepHypRef Expression
  1 df-cn 20971 . . 3 Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})
  21elmpt2cl 6841 . 2 (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
  3 iscn.1 . . . 4 𝑋 = 𝐽
  43toptopon 20662 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
  5 iscn.2 . . . 4 𝑌 = 𝐾
  65toptopon 20662 . . 3 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
  7 iscn 20979 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
  84, 6, 7syl2anb 496 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
  92, 8biadan2 673 1 (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
   Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2908  {crab 2912  ∪ cuni 4409  ◡ccnv 5083   “ cima 5087  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615   ↑𝑚 cmap 7817  Topctop 20638  TopOnctopon 20655   Cn ccn 20968 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-top 20639  df-topon 20656  df-cn 20971 This theorem is referenced by:  cntop1  20984  cntop2  20985  cnf  20990  cnima  21009  cnco  21010  ptpjcn  21354
   Copyright terms: Public domain W3C validator
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a formal proof in the Metamath Proof Explorer system. It defines and proves a theorem about continuous functions in topology. The theorem states the conditions under which a function \( F \) is considered continuous between two topological spaces \( J \) and \( K \). The text does not contain a question or an answer in the conventional sense; it is a structured proof with hypotheses, assertions, and steps.
  
  Label: no-question