Theorem foeq1

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

	Ref	Expression
Assertion	foeq1	|- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> 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	eqeq1d	|- ( F = G -> ( ran F = B <-> ran G = B ) )
4	1 3	anbi12d	|- ( F = G -> ( ( F Fn A /\ ran F = B ) <-> ( G Fn A /\ ran G = B ) ) )
5		df-fo	|- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6		df-fo	|- ( G : A -onto-> B <-> ( G Fn A /\ ran G = B ) )
7	4 5 6	3bitr4g	|- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )