Theorem compsscnvlem 9192

Description: Lemma for compsscnv 9193. (Contributed by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
compsscnvlem	|- ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( y e. ~P A /\ x = ( A \ y ) ) )

Distinct variable group:   x, y, A

Proof of Theorem compsscnvlem

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> y = ( A \ x ) )
2		difss 3737	. . . 4 |- ( A \ x ) C_ A
3	1, 2	syl6eqss 3655	. . 3 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> y C_ A )
4		selpw 4165	. . 3 |- ( y e. ~P A <-> y C_ A )
5	3, 4	sylibr 224	. 2 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> y e. ~P A )
6	1	difeq2d 3728	. . 3 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( A \ y ) = ( A \ ( A \ x ) ) )
7		elpwi 4168	. . . . 5 |- ( x e. ~P A -> x C_ A )
8	7	adantr 481	. . . 4 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> x C_ A )
9		dfss4 3858	. . . 4 |- ( x C_ A <-> ( A \ ( A \ x ) ) = x )
10	8, 9	sylib 208	. . 3 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( A \ ( A \ x ) ) = x )
11	6, 10	eqtr2d 2657	. 2 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> x = ( A \ y ) )
12	5, 11	jca 554	1 |- ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( y e. ~P A /\ x = ( A \ y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   ~Pcpw 4158

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-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  compsscnv  9193