Theorem feq1

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

	Ref	Expression
Assertion	feq1	|- ( F = G -> ( F : A --> B <-> G : A --> B ) )

Proof

Step	Hyp	Ref	Expression
1		fneq1	|- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq	|- ( F = G -> ran F = ran G )
3	2	sseq1d	|- ( F = G -> ( ran F C_ B <-> ran G C_ B ) )
4	1 3	anbi12d	|- ( F = G -> ( ( F Fn A /\ ran F C_ B ) <-> ( G Fn A /\ ran G C_ B ) ) )
5		df-f	|- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f	|- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
7	4 5 6	3bitr4g	|- ( F = G -> ( F : A --> B <-> G : A --> B ) )