Theorem f1eq2 5088

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 5031	. . 3 |- ( A = B -> ( F : A --> C <-> F : B --> C ) )
2	1	anbi1d 438	. 2 |- ( A = B -> ( ( F : A --> C /\ Fun `' F ) <-> ( F : B --> C /\ Fun `' F ) ) )
3		df-f1 4907	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
4		df-f1 4907	. 2 |- ( F : B -1-1-> C <-> ( F : B --> C /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 212	1 |- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   `'ccnv 4344   Fun wfun 4896   -->wf 4898   -1-1->wf1 4899

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-fn 4905  df-f 4906  df-f1 4907

This theorem is referenced by:  f1oeq2  5118  f1eq123d  5121  brdomg  6229