Theorem feq2 5351

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 5307	. . 3 |- ( A = B -> ( F Fn A <-> F Fn B ) )
2	1	anbi1d 465	. 2 |- ( A = B -> ( ( F Fn A /\ ran F C_ C ) <-> ( F Fn B /\ ran F C_ C ) ) )
3		df-f 5222	. 2 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
4		df-f 5222	. 2 |- ( F : B --> C <-> ( F Fn B /\ ran F C_ C ) )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( F : A --> C <-> F : B --> C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   C_ wss 3131   ran crn 4629   Fn wfn 5213   -->wf 5214

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-fn 5221  df-f 5222

This theorem is referenced by:  feq23  5353  feq2d  5355  feq2i  5361  f00  5409  f0dom0  5411  f1eq2  5419  fressnfv  5706  tfrcllemsucfn  6357  tfrcllemsucaccv  6358  tfrcllembxssdm  6360  tfrcllembfn  6361  tfrcllemaccex  6365  tfrcllemres  6366  tfrcldm  6367  tfrcl  6368  mapvalg  6661  map0g  6691  ac6sfi  6901  isomni  7137  ismkv  7154  iswomni  7166