Theorem f1eq3 5759

Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Assertion

Ref	Expression
f1eq3	|- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )

Proof of Theorem f1eq3

Step	Hyp	Ref	Expression
1		feq3 5694	. . 3 |- ( A = B -> ( F : C --> A <-> F : C --> B ) )
2	1	anbi1d 716	. 2 |- ( A = B -> ( ( F : C --> A /\ Fun `' F ) <-> ( F : C --> B /\ Fun `' F ) ) )
3		df-f1 5566	. 2 |- ( F : C -1-1-> A <-> ( F : C --> A /\ Fun `' F ) )
4		df-f1 5566	. 2 |- ( F : C -1-1-> B <-> ( F : C --> B /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 296	1 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1448   `'ccnv 4811   Fun wfun 5555   -->wf 5557   -1-1->wf1 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-in 3379  df-ss 3386  df-f 5565  df-f1 5566

This theorem is referenced by:  f1oeq3  5790  f1eq123d  5792  tposf12  6985  brdomg  7566  pwfseq  9076  f1linds  19394  isuslgra  25082  isusgra  25083  isusgra0  25086  usgraop  25089  cusgraexilem2  25207  diaf1oN  34700  isusgrs  39343  usgr1vr  39431  usgrexi  39608