Theorem f1eq2 5399

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 5331	. . 3 |- ( A = B -> ( F : A --> C <-> F : B --> C ) )
2	1	anbi1d 462	. 2 |- ( A = B -> ( ( F : A --> C /\ Fun `' F ) <-> ( F : B --> C /\ Fun `' F ) ) )
3		df-f1 5203	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
4		df-f1 5203	. 2 |- ( F : B -1-1-> C <-> ( F : B --> C /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   `'ccnv 4610   Fun wfun 5192   -->wf 5194   -1-1->wf1 5195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-fn 5201  df-f 5202  df-f1 5203

This theorem is referenced by:  f1oeq2  5432  f1eq123d  5435  brdomg  6726  ennnfonelemen  12376