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Metamath Proof Explorer

Theorem compsscnvlem

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

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

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( x e. ~P A /\ y = ( A \ x ) ) -> y = ( A \ x ) )
2 difss 
 |-  ( A \ x ) C_ A
3 1 2 eqsstrdi 
 |-  ( ( x e. ~P A /\ y = ( A \ x ) ) -> y C_ A )
4 velpw 
 |-  ( y e. ~P A <-> y C_ A )
5 3 4 sylibr 
 |-  ( ( x e. ~P A /\ y = ( A \ x ) ) -> y e. ~P A )
6 1 difeq2d 
 |-  ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( A \ y ) = ( A \ ( A \ x ) ) )
7 elpwi 
 |-  ( x e. ~P A -> x C_ A )
8 7 adantr 
 |-  ( ( x e. ~P A /\ y = ( A \ x ) ) -> x C_ A )
9 dfss4 
 |-  ( x C_ A <-> ( A \ ( A \ x ) ) = x )
10 8 9 sylib 
 |-  ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( A \ ( A \ x ) ) = x )
11 6 10 eqtr2d 
 |-  ( ( x e. ~P A /\ y = ( A \ x ) ) -> x = ( A \ y ) )
12 5 11 jca 
 |-  ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( y e. ~P A /\ x = ( A \ y ) ) )