Theorem fpwwe2lem1 9453

Description: Lemma for fpwwe2 9465. (Contributed by Mario Carneiro, 15-May-2015.)

Hypothesis

Ref	Expression
fpwwe2.1	|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }

Assertion

Ref	Expression
fpwwe2lem1	|- W C_ ( ~P A X. ~P ( A X. A ) )

Distinct variable groups:   y, u, r, x, F   A, r, x   W, r, u, x, y

Allowed substitution hints:   A( y, u)

Proof of Theorem fpwwe2lem1

Step	Hyp	Ref	Expression
1		simpll 790	. . . . 5 |- ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) -> x C_ A )
2		selpw 4165	. . . . 5 |- ( x e. ~P A <-> x C_ A )
3	1, 2	sylibr 224	. . . 4 |- ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) -> x e. ~P A )
4		simplr 792	. . . . . 6 |- ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) -> r C_ ( x X. x ) )
5		xpss12 5225	. . . . . . 7 |- ( ( x C_ A /\ x C_ A ) -> ( x X. x ) C_ ( A X. A ) )
6	1, 1, 5	syl2anc 693	. . . . . 6 |- ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) -> ( x X. x ) C_ ( A X. A ) )
7	4, 6	sstrd 3613	. . . . 5 |- ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) -> r C_ ( A X. A ) )
8		selpw 4165	. . . . 5 |- ( r e. ~P ( A X. A ) <-> r C_ ( A X. A ) )
9	7, 8	sylibr 224	. . . 4 |- ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) -> r e. ~P ( A X. A ) )
10	3, 9	jca 554	. . 3 |- ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) -> ( x e. ~P A /\ r e. ~P ( A X. A ) ) )
11	10	ssopab2i 5003	. 2 |- { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) } C_ { <. x , r >. | ( x e. ~P A /\ r e. ~P ( A X. A ) ) }
12		fpwwe2.1	. 2 |- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }
13		df-xp 5120	. 2 |- ( ~P A X. ~P ( A X. A ) ) = { <. x , r >. | ( x e. ~P A /\ r e. ~P ( A X. A ) ) }
14	11, 12, 13	3sstr4i 3644	1 |- W C_ ( ~P A X. ~P ( A X. A ) )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   [.wsbc 3435   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   {copab 4712   We wwe 5072   X. cxp 5112   `'ccnv 5113   "cima 5117  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-opab 4713  df-xp 5120

This theorem is referenced by: (None)