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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 453 . 2 (𝐴 = 𝐵 → ((𝐹:𝐴𝐶 ∧ Fun 𝐹) ↔ (𝐹:𝐵𝐶 ∧ Fun 𝐹)))
3 df-f1 4931 . 2 (𝐹:𝐴1-1𝐶 ↔ (𝐹:𝐴𝐶 ∧ Fun 𝐹))
4 df-f1 4931 . 2 (𝐹:𝐵1-1𝐶 ↔ (𝐹:𝐵𝐶 ∧ Fun 𝐹))
52, 3, 43bitr4g 221 1 (𝐴 = 𝐵 → (𝐹:𝐴1-1𝐶𝐹:𝐵1-1𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285  ◡ccnv 4364  Fun wfun 4920  ⟶wf 4922  –1-1→wf1 4923 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-fn 4929  df-f 4930  df-f1 4931 This theorem is referenced by:  f1oeq2  5143  f1eq123d  5146  brdomg  6288
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