Theorem fndmdifeq0

Description: The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015)

	Ref	Expression
Assertion	fndmdifeq0	|- ( ( F Fn A /\ G Fn A ) -> ( dom ( F \ G ) = (/) <-> F = G ) )

Proof

Step	Hyp	Ref	Expression
1		fndmdif	|- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
2	1	eqeq1d	|- ( ( F Fn A /\ G Fn A ) -> ( dom ( F \ G ) = (/) <-> { x e. A | ( F ` x ) =/= ( G ` x ) } = (/) ) )
3		rabeq0	|- ( { x e. A | ( F ` x ) =/= ( G ` x ) } = (/) <-> A. x e. A -. ( F ` x ) =/= ( G ` x ) )
4		nne	|- ( -. ( F ` x ) =/= ( G ` x ) <-> ( F ` x ) = ( G ` x ) )
5	4	ralbii	|- ( A. x e. A -. ( F ` x ) =/= ( G ` x ) <-> A. x e. A ( F ` x ) = ( G ` x ) )
6	3 5	bitri	|- ( { x e. A | ( F ` x ) =/= ( G ` x ) } = (/) <-> A. x e. A ( F ` x ) = ( G ` x ) )
7		eqfnfv	|- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
8	6 7	bitr4id	|- ( ( F Fn A /\ G Fn A ) -> ( { x e. A | ( F ` x ) =/= ( G ` x ) } = (/) <-> F = G ) )
9	2 8	bitrd	|- ( ( F Fn A /\ G Fn A ) -> ( dom ( F \ G ) = (/) <-> F = G ) )