Theorem f1eq3 5228

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 5162	. . 3 |- ( A = B -> ( F : C --> A <-> F : C --> B ) )
2	1	anbi1d 454	. 2 |- ( A = B -> ( ( F : C --> A /\ Fun `' F ) <-> ( F : C --> B /\ Fun `' F ) ) )
3		df-f1 5035	. 2 |- ( F : C -1-1-> A <-> ( F : C --> A /\ Fun `' F ) )
4		df-f1 5035	. 2 |- ( F : C -1-1-> B <-> ( F : C --> B /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   `'ccnv 4453   Fun wfun 5024   -->wf 5026   -1-1->wf1 5027

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3008  df-ss 3015  df-f 5034  df-f1 5035

This theorem is referenced by:  f1oeq3  5261  f1eq123d  5263  tposf12  6050  brdomg  6521