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Theorem f1eq3 6800
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 6717 . . 3 (𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))
21anbi1d 631 . 2 (𝐴 = 𝐵 → ((𝐹:𝐶𝐴 ∧ Fun 𝐹) ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹)))
3 df-f1 6565 . 2 (𝐹:𝐶1-1𝐴 ↔ (𝐹:𝐶𝐴 ∧ Fun 𝐹))
4 df-f1 6565 . 2 (𝐹:𝐶1-1𝐵 ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹))
52, 3, 43bitr4g 314 1 (𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  ccnv 5683  Fun wfun 6554  wf 6556  1-1wf1 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2728  df-ss 3967  df-f 6564  df-f1 6565
This theorem is referenced by:  f1oeq3  6837  f1eq123d  6839  tposf12  8277  brdom2g  8997  brdomgOLD  8999  1sdom2dom  9284  pwfseq  10705  f1linds  21846  isusgrs  29174  usgrstrrepe  29253  usgrexilem  29458  tocycval  33129  diaf1oN  41133  f1cof1b  47094
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