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Theorem f1eq3 5114
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1eq3 (𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))

Proof of Theorem f1eq3
StepHypRef Expression
1 feq3 5057 . . 3 (𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))
21anbi1d 453 . 2 (𝐴 = 𝐵 → ((𝐹:𝐶𝐴 ∧ Fun 𝐹) ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹)))
3 df-f1 4931 . 2 (𝐹:𝐶1-1𝐴 ↔ (𝐹:𝐶𝐴 ∧ Fun 𝐹))
4 df-f1 4931 . 2 (𝐹:𝐶1-1𝐵 ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹))
52, 3, 43bitr4g 221 1 (𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  ccnv 4364  Fun wfun 4920  wf 4922  1-1wf1 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-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987  df-f 4930  df-f1 4931
This theorem is referenced by:  f1oeq3  5144  f1eq123d  5146  tposf12  5912  brdomg  6288
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