Theorem f1eq3 5500

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 5430	. . 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 5295	. 2 |- ( F : C -1-1-> A <-> ( F : C --> A /\ Fun `' F ) )
4		df-f1 5295	. 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 1373   `'ccnv 4692   Fun wfun 5284   -->wf 5286   -1-1->wf1 5287

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187  df-f 5294  df-f1 5295

This theorem is referenced by:  f1oeq3  5534  f1eq123d  5536  tposf12  6378  brdom2g  6859  brdomg  6860