Theorem isf34lem2

Description: Lemma for isfin3-4 . (Contributed by Stefan O'Rear, 7-Nov-2014)

	Ref	Expression
Hypothesis	compss.a	|- F = ( x e. ~P A |-> ( A \ x ) )
Assertion	isf34lem2	|- ( A e. V -> F : ~P A --> ~P A )

Proof

Step	Hyp	Ref	Expression
1		compss.a	|- F = ( x e. ~P A |-> ( A \ x ) )
2		difss	|- ( A \ x ) C_ A
3		elpw2g	|- ( A e. V -> ( ( A \ x ) e. ~P A <-> ( A \ x ) C_ A ) )
4	2 3	mpbiri	|- ( A e. V -> ( A \ x ) e. ~P A )
5	4	adantr	|- ( ( A e. V /\ x e. ~P A ) -> ( A \ x ) e. ~P A )
6	5 1	fmptd	|- ( A e. V -> F : ~P A --> ~P A )