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Theorem f1eq3 6572
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 6488 . . 3 (𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))
21anbi1d 633 . 2 (𝐴 = 𝐵 → ((𝐹:𝐶𝐴 ∧ Fun 𝐹) ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹)))
3 df-f1 6345 . 2 (𝐹:𝐶1-1𝐴 ↔ (𝐹:𝐶𝐴 ∧ Fun 𝐹))
4 df-f1 6345 . 2 (𝐹:𝐶1-1𝐵 ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹))
52, 3, 43bitr4g 317 1 (𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  ccnv 5525  Fun wfun 6334  wf 6336  1-1wf1 6337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3401  df-in 3851  df-ss 3861  df-f 6344  df-f1 6345
This theorem is referenced by:  f1oeq3  6611  f1eq123d  6613  tposf12  7949  brdomg  8568  pwfseq  10167  f1linds  20644  isusgrs  27104  usgrstrrepe  27180  usgrexilem  27385  tocycval  30955  diaf1oN  38790  f1cof1b  44131
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