Theorem f1eq2 5212

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 5146	. . 3 |- ( A = B -> ( F : A --> C <-> F : B --> C ) )
2	1	anbi1d 453	. 2 |- ( A = B -> ( ( F : A --> C /\ Fun `' F ) <-> ( F : B --> C /\ Fun `' F ) ) )
3		df-f1 5020	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
4		df-f1 5020	. 2 |- ( F : B -1-1-> C <-> ( F : B --> C /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 221	1 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   `'ccnv 4437   Fun wfun 5009   -->wf 5011   -1-1->wf1 5012

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-fn 5018  df-f 5019  df-f1 5020

This theorem is referenced by:  f1oeq2  5245  f1eq123d  5248  brdomg  6465