MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1eq3 Structured version   Visualization version   GIF version

Theorem f1eq3 5995
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 5926 . . 3 (𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))
21anbi1d 736 . 2 (𝐴 = 𝐵 → ((𝐹:𝐶𝐴 ∧ Fun 𝐹) ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹)))
3 df-f1 5794 . 2 (𝐹:𝐶1-1𝐴 ↔ (𝐹:𝐶𝐴 ∧ Fun 𝐹))
4 df-f1 5794 . 2 (𝐹:𝐶1-1𝐵 ↔ (𝐹:𝐶𝐵 ∧ Fun 𝐹))
52, 3, 43bitr4g 301 1 (𝐴 = 𝐵 → (𝐹:𝐶1-1𝐴𝐹:𝐶1-1𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  ccnv 5026  Fun wfun 5783  wf 5785  1-1wf1 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-in 3546  df-ss 3553  df-f 5793  df-f1 5794
This theorem is referenced by:  f1oeq3  6026  f1eq123d  6028  tposf12  7241  brdomg  7828  pwfseq  9342  f1linds  19930  isuslgra  25665  isusgra  25666  isusgra0  25669  usgraop  25672  cusgraexilem2  25789  diaf1oN  35220  isusgrs  40367  usgrexi  40642
  Copyright terms: Public domain W3C validator