Theorem fnimadisj

Description: A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007)

	Ref	Expression
Assertion	fnimadisj	|- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( F " C ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		fndm	|- ( F Fn A -> dom F = A )
2	1	ineq1d	|- ( F Fn A -> ( dom F i^i C ) = ( A i^i C ) )
3	2	eqeq1d	|- ( F Fn A -> ( ( dom F i^i C ) = (/) <-> ( A i^i C ) = (/) ) )
4	3	biimpar	|- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( dom F i^i C ) = (/) )
5		imadisj	|- ( ( F " C ) = (/) <-> ( dom F i^i C ) = (/) )
6	4 5	sylibr	|- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( F " C ) = (/) )