Theorem f1eq3 5528

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 5458	. . 3 |- ( A = B -> ( F : C --> A <-> F : C --> B ) )
2	1	anbi1d 465	. 2 |- ( A = B -> ( ( F : C --> A /\ Fun `' F ) <-> ( F : C --> B /\ Fun `' F ) ) )
3		df-f1 5323	. 2 |- ( F : C -1-1-> A <-> ( F : C --> A /\ Fun `' F ) )
4		df-f1 5323	. 2 |- ( F : C -1-1-> B <-> ( F : C --> B /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   `'ccnv 4718   Fun wfun 5312   -->wf 5314   -1-1->wf1 5315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-f 5322  df-f1 5323

This theorem is referenced by:  f1oeq3  5562  f1eq123d  5564  tposf12  6415  brdom2g  6896  brdomg  6897  usgrstrrepeen  16029