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Theorem f1eq2 5113
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1eq2 (𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))

Proof of Theorem f1eq2
StepHypRef Expression
1 feq2 5056 . . 3 (𝐴 = 𝐵 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))
21anbi1d 446 . 2 (𝐴 = 𝐵 → ((𝐹:𝐴𝐶 ∧ Fun 𝐹) ↔ (𝐹:𝐵𝐶 ∧ Fun 𝐹)))
3 df-f1 4932 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
4 df-f1 4932 . 2 (𝐹:𝐵1-1𝐶 ↔ (𝐹:𝐵𝐶 ∧ Fun 𝐹))
52, 3, 43bitr4g 216 1 (𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  ccnv 4369  Fun wfun 4921  wf 4923  1-1wf1 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1350  ax-gen 1352  ax-4 1414  ax-17 1433  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-cleq 2047  df-fn 4930  df-f 4931  df-f1 4932
This theorem is referenced by:  f1oeq2  5143  f1eq123d  5146  brdomg  6257
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