Theorem f1eq2 5574

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

Assertion

Ref	Expression
f1eq2	|- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )

Proof of Theorem f1eq2

Step	Hyp	Ref	Expression
1		feq2 5497	. . 3 |- ( A = B -> ( F : A --> C <-> F : B --> C ) )
2	1	anbi1d 465	. 2 |- ( A = B -> ( ( F : A --> C /\ Fun `' F ) <-> ( F : B --> C /\ Fun `' F ) ) )
3		df-f1 5362	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
4		df-f1 5362	. 2 |- ( F : B -1-1-> C <-> ( F : B --> C /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   `'ccnv 4753   Fun wfun 5351   -->wf 5353   -1-1->wf1 5354

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-fn 5360  df-f 5361  df-f1 5362

This theorem is referenced by:  f1oeq2  5608  f1eq123d  5611  f10d  5655  brdom2g  6997  brdomg  6998  dom1o  7082  ennnfonelemen  13256  ausgrusgrben  16289  usgr0  16360  uspgr1edc  16361