Theorem f1eq3 6098

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 6028	. . 3 |- ( A = B -> ( F : C --> A <-> F : C --> B ) )
2	1	anbi1d 741	. 2 |- ( A = B -> ( ( F : C --> A /\ Fun `' F ) <-> ( F : C --> B /\ Fun `' F ) ) )
3		df-f1 5893	. 2 |- ( F : C -1-1-> A <-> ( F : C --> A /\ Fun `' F ) )
4		df-f1 5893	. 2 |- ( F : C -1-1-> B <-> ( F : C --> B /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 303	1 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   `'ccnv 5113   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-f 5892  df-f1 5893

This theorem is referenced by:  f1oeq3  6129  f1eq123d  6131  tposf12  7377  brdomg  7965  pwfseq  9486  f1linds  20164  isusgrs  26051  usgrstrrepe  26127  usgrexilem  26336  diaf1oN  36419