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Theorem f1eq3 6335
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 6261 . . 3 (𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))
21anbi1d 623 . 2 (𝐴 = 𝐵 → ((𝐹:𝐶𝐴 ∧ Fun 𝐹) ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹)))
3 df-f1 6128 . 2 (𝐹:𝐶1-1𝐴 ↔ (𝐹:𝐶𝐴 ∧ Fun 𝐹))
4 df-f1 6128 . 2 (𝐹:𝐶1-1𝐵 ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹))
52, 3, 43bitr4g 306 1 (𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  ccnv 5341  Fun wfun 6117  wf 6119  1-1wf1 6120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-in 3805  df-ss 3812  df-f 6127  df-f1 6128
This theorem is referenced by:  f1oeq3  6369  f1eq123d  6371  tposf12  7642  brdomg  8232  pwfseq  9801  f1linds  20531  isusgrs  26455  usgrstrrepe  26532  usgrexilem  26738  diaf1oN  37198
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