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Metamath Proof Explorer

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 eqtr3id 
 |-  ( ( A e. V /\ X C_ ~P A ) -> ( F " ( F " X ) ) = X )