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  Theorem coaval 17387
   Description: Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
  Hypotheses
  Ref Expression
  homdmcoa.o · = (compa𝐶)
  homdmcoa.h 𝐻 = (Homa𝐶)
  homdmcoa.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
  homdmcoa.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
  coaval.x = (comp‘𝐶)
  Assertion
  Ref Expression
  coaval (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
  
  Proof of Theorem coaval
  Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
  StepHypRef Expression
  1 homdmcoa.o . . 3 · = (compa𝐶)
  2 eqid 2759 . . 3 (Arrow‘𝐶) = (Arrow‘𝐶)
  3 coaval.x . . 3 = (comp‘𝐶)
  41, 2, 3coafval 17383 . 2 · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
  5 homdmcoa.h . . . . 5 𝐻 = (Homa𝐶)
  62, 5homarw 17365 . . . 4 (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
  7 homdmcoa.g . . . 4 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
  86, 7sseldi 3891 . . 3 (𝜑𝐺 ∈ (Arrow‘𝐶))
  9 fveqeq2 6668 . . . 4 ( = 𝐹 → ((coda) = (doma𝑔) ↔ (coda𝐹) = (doma𝑔)))
  102, 5homarw 17365 . . . . 5 (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
  11 homdmcoa.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
  1211adantr 485 . . . . 5 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (𝑋𝐻𝑌))
  1310, 12sseldi 3891 . . . 4 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
  145homacd 17360 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
  1512, 14syl 17 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝐹) = 𝑌)
  16 simpr 489 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
  1716fveq2d 6663 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝑔) = (doma𝐺))
  187adantr 485 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
  195homadm 17359 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (doma𝐺) = 𝑌)
  2018, 19syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐺) = 𝑌)
  2117, 20eqtrd 2794 . . . . 5 ((𝜑𝑔 = 𝐺) → (doma𝑔) = 𝑌)
  2215, 21eqtr4d 2797 . . . 4 ((𝜑𝑔 = 𝐺) → (coda𝐹) = (doma𝑔))
  239, 13, 22elrabd 3605 . . 3 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)})
  24 otex 5326 . . . 4 ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ ∈ V
  2524a1i 11 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ ∈ V)
  26 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
  2726fveq2d 6663 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = (doma𝐹))
  285homadm 17359 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
  2912, 28syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐹) = 𝑋)
  3029adantrr 717 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝐹) = 𝑋)
  3127, 30eqtrd 2794 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = 𝑋)
  3216fveq2d 6663 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝑔) = (coda𝐺))
  335homacd 17360 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (coda𝐺) = 𝑍)
  3418, 33syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝐺) = 𝑍)
  3532, 34eqtrd 2794 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝑔) = 𝑍)
  3635adantrr 717 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (coda𝑔) = 𝑍)
  3721adantrr 717 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑔) = 𝑌)
  3831, 37opeq12d 4772 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (doma𝑔)⟩ = ⟨𝑋, 𝑌⟩)
  3938, 36oveq12d 7169 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔)) = (⟨𝑋, 𝑌 𝑍))
  40 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
  4140fveq2d 6663 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
  4226fveq2d 6663 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
  4339, 41, 42oveq123d 7172 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))
  4431, 36, 43oteq123d 4779 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
  458, 23, 25, 44ovmpodv2 7304 . 2 (𝜑 → ( · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩))
  464, 45mpi 20 1 (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
   Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {crab 3075  Vcvv 3410  ⟨cop 4529  ⟨cotp 4531  ‘cfv 6336  (class class class)co 7151   ∈ cmpo 7153  2nd c2nd 7693  compcco 16628  domacdoma 17339  codaccoda 17340  Arrowcarw 17341  Homachoma 17342  compaccoa 17373 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-ot 4532  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-doma 17343  df-coda 17344  df-homa 17345  df-arw 17346  df-coa 17375 This theorem is referenced by:  coa2  17388  coahom  17389  arwlid  17391  arwrid  17392  arwass  17393
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