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Theorem compsscnvlem 8251
Description: Lemma for compsscnv 8252. (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
StepHypRef Expression
1 simpr 449 . . . 4  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  =  ( A  \  x ) )
2 difss 3475 . . . 4  |-  ( A 
\  x )  C_  A
31, 2syl6eqss 3399 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  C_  A
)
4 vex 2960 . . . 4  |-  y  e. 
_V
54elpw 3806 . . 3  |-  ( y  e.  ~P A  <->  y  C_  A )
63, 5sylibr 205 . 2  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  e.  ~P A )
71difeq2d 3466 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( A  \ 
y )  =  ( A  \  ( A 
\  x ) ) )
8 elpwi 3808 . . . . 5  |-  ( x  e.  ~P A  ->  x  C_  A )
98adantr 453 . . . 4  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  x  C_  A
)
10 dfss4 3576 . . . 4  |-  ( x 
C_  A  <->  ( A  \  ( A  \  x
) )  =  x )
119, 10sylib 190 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( A  \ 
( A  \  x
) )  =  x )
127, 11eqtr2d 2470 . 2  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  x  =  ( A  \  y ) )
136, 12jca 520 1  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3318    C_ wss 3321   ~Pcpw 3800
This theorem is referenced by:  compsscnv  8252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rab 2715  df-v 2959  df-dif 3324  df-in 3328  df-ss 3335  df-pw 3802
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