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Theorem compsscnvlem 8751
Description: Lemma for compsscnv 8752. (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 461 . . . 4  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  =  ( A  \  x ) )
2 difss 3631 . . . 4  |-  ( A 
\  x )  C_  A
31, 2syl6eqss 3554 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  C_  A
)
4 selpw 4017 . . 3  |-  ( y  e.  ~P A  <->  y  C_  A )
53, 4sylibr 212 . 2  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  e.  ~P A )
61difeq2d 3622 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( A  \ 
y )  =  ( A  \  ( A 
\  x ) ) )
7 elpwi 4019 . . . . 5  |-  ( x  e.  ~P A  ->  x  C_  A )
87adantr 465 . . . 4  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  x  C_  A
)
9 dfss4 3732 . . . 4  |-  ( x 
C_  A  <->  ( A  \  ( A  \  x
) )  =  x )
108, 9sylib 196 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( A  \ 
( A  \  x
) )  =  x )
116, 10eqtr2d 2509 . 2  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  x  =  ( A  \  y ) )
125, 11jca 532 1  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   ~Pcpw 4010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-pw 4012
This theorem is referenced by:  compsscnv  8752
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