Theorem isf34lem3

Description: Lemma for isfin3-4 . (Contributed by Stefan O'Rear, 7-Nov-2014) (Revised by Mario Carneiro, 17-May-2015)

	Ref	Expression
Hypothesis	compss.a	|- F = ( x e. ~P A |-> ( A \ x ) )
Assertion	isf34lem3	|- ( ( A e. V /\ X C_ ~P A ) -> ( F " ( F " X ) ) = X )

Proof

Step	Hyp	Ref	Expression
1		compss.a	|- F = ( x e. ~P A |-> ( A \ x ) )
2	1	compsscnv	|- `' F = F
3	2	imaeq1i	|- ( `' F " ( F " X ) ) = ( F " ( F " X ) )
4	1	compssiso	|- ( A e. V -> F Isom [C.] , `' [C.] ( ~P A , ~P A ) )
5		isof1o	|- ( F Isom [C.] , `' [C.] ( ~P A , ~P A ) -> F : ~P A -1-1-onto-> ~P A )
6		f1of1	|- ( F : ~P A -1-1-onto-> ~P A -> F : ~P A -1-1-> ~P A )
7	4 5 6	3syl	|- ( A e. V -> F : ~P A -1-1-> ~P A )
8		f1imacnv	|- ( ( F : ~P A -1-1-> ~P A /\ X C_ ~P A ) -> ( `' F " ( F " X ) ) = X )
9	7 8	sylan	|- ( ( A e. V /\ X C_ ~P A ) -> ( `' F " ( F " X ) ) = X )
10	3 9	syl5eqr	|- ( ( A e. V /\ X C_ ~P A ) -> ( F " ( F " X ) ) = X )