Theorem feq2 5051

Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
feq2	|- ( A = B -> ( F : A --> C <-> F : B --> C ) )

Proof of Theorem feq2

Step	Hyp	Ref	Expression
1		fneq2 5008	. . 3 |- ( A = B -> ( F Fn A <-> F Fn B ) )
2	1	anbi1d 452	. 2 |- ( A = B -> ( ( F Fn A /\ ran F C_ C ) <-> ( F Fn B /\ ran F C_ C ) ) )
3		df-f 4926	. 2 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
4		df-f 4926	. 2 |- ( F : B --> C <-> ( F Fn B /\ ran F C_ C ) )
5	2, 3, 4	3bitr4g 221	1 |- ( A = B -> ( F : A --> C <-> F : B --> C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   C_ wss 2973   ran crn 4364   Fn wfn 4917   -->wf 4918

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 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-fn 4925  df-f 4926

This theorem is referenced by:  feq23  5053  feq2d  5055  feq2i  5060  f00  5101  f1eq2  5108  fressnfv  5371  ac6sfi  6379