text: # Metamath Proof Explorer
  
  ## Theorem eqfnfv2f
  
  Description: Equality of functions is determined by their values. Special case of Exercise 4 of TakeutiZaring p. 28 (with domain equality omitted). This version of eqfnfv uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004)
  
  Ref Expression
  Hypotheses eqfnfv2f.1 𝑥 𝐹
  eqfnfv2f.2 𝑥 𝐺
  Assertion eqfnfv2f ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ∀ 𝑥𝐴 ( 𝐹𝑥 ) = ( 𝐺𝑥 ) ) )
  
  ### Proof
  
  Step Hyp Ref Expression
  1 eqfnfv2f.1 𝑥 𝐹
  2 eqfnfv2f.2 𝑥 𝐺
  3 eqfnfv ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ∀ 𝑧𝐴 ( 𝐹𝑧 ) = ( 𝐺𝑧 ) ) )
  4 nfcv 𝑥 𝑧
  5 1 4 nffv 𝑥 ( 𝐹𝑧 )
  6 2 4 nffv 𝑥 ( 𝐺𝑧 )
  7 5 6 nfeq 𝑥 ( 𝐹𝑧 ) = ( 𝐺𝑧 )
  8 nfv 𝑧 ( 𝐹𝑥 ) = ( 𝐺𝑥 )
  9 fveq2 ( 𝑧 = 𝑥 → ( 𝐹𝑧 ) = ( 𝐹𝑥 ) )
  10 fveq2 ( 𝑧 = 𝑥 → ( 𝐺𝑧 ) = ( 𝐺𝑥 ) )
  11 9 10 eqeq12d ( 𝑧 = 𝑥 → ( ( 𝐹𝑧 ) = ( 𝐺𝑧 ) ↔ ( 𝐹𝑥 ) = ( 𝐺𝑥 ) ) )
  12 7 8 11 cbvralw ( ∀ 𝑧𝐴 ( 𝐹𝑧 ) = ( 𝐺𝑧 ) ↔ ∀ 𝑥𝐴 ( 𝐹𝑥 ) = ( 𝐺𝑥 ) )
  13 3 12 syl6bb ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ∀ 𝑥𝐴 ( 𝐹𝑥 ) = ( 𝐺𝑥 ) ) )
explanation: The provided text is a formal proof in the Metamath Proof Explorer, which is a project aimed at developing a comprehensive database of mathematical proofs. The text describes a theorem (eqfnfv2f) that deals with the equality of functions based on their values. It includes hypotheses, an assertion, and a detailed step-by-step proof. The document does not contain a question or an answer to a question.
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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distilabel_metadata.role: system
distilabel_metadata.content: # Metamath Proof Explorer
  
  ## Theorem eqfnfv2f
  
  Description: Equality of functions is determined by their values. Special case of Exercise 4 of TakeutiZaring p. 28 (with domain equality omitted). This version of eqfnfv uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004)
  
  Ref Expression
  Hypotheses eqfnfv2f.1 𝑥 𝐹
  eqfnfv2f.2 𝑥 𝐺
  Assertion eqfnfv2f ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ∀ 𝑥𝐴 ( 𝐹𝑥 ) = ( 𝐺𝑥 ) ) )
  
  ### Proof
  
  Step Hyp Ref Expression
  1 eqfnfv2f.1 𝑥 𝐹
  2 eqfnfv2f.2 𝑥 𝐺
  3 eqfnfv ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ∀ 𝑧𝐴 ( 𝐹𝑧 ) = ( 𝐺𝑧 ) ) )
  4 nfcv 𝑥 𝑧
  5 1 4 nffv 𝑥 ( 𝐹𝑧 )
  6 2 4 nffv 𝑥 ( 𝐺𝑧 )
  7 5 6 nfeq 𝑥 ( 𝐹𝑧 ) = ( 𝐺𝑧 )
  8 nfv 𝑧 ( 𝐹𝑥 ) = ( 𝐺𝑥 )
  9 fveq2 ( 𝑧 = 𝑥 → ( 𝐹𝑧 ) = ( 𝐹𝑥 ) )
  10 fveq2 ( 𝑧 = 𝑥 → ( 𝐺𝑧 ) = ( 𝐺𝑥 ) )
  11 9 10 eqeq12d ( 𝑧 = 𝑥 → ( ( 𝐹𝑧 ) = ( 𝐺𝑧 ) ↔ ( 𝐹𝑥 ) = ( 𝐺𝑥 ) ) )
  12 7 8 11 cbvralw ( ∀ 𝑧𝐴 ( 𝐹𝑧 ) = ( 𝐺𝑧 ) ↔ ∀ 𝑥𝐴 ( 𝐹𝑥 ) = ( 𝐺𝑥 ) )
  13 3 12 syl6bb ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ∀ 𝑥𝐴 ( 𝐹𝑥 ) = ( 𝐺𝑥 ) ) )
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a formal proof in the Metamath Proof Explorer, which is a project aimed at developing a comprehensive database of mathematical proofs. The text describes a theorem (eqfnfv2f) that deals with the equality of functions based on their values. It includes hypotheses, an assertion, and a detailed step-by-step proof. The document does not contain a question or an answer to a question.
  
  Label: no-question