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Theorem compsscnvlem 8741
Description: Lemma for compsscnv 8742. (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 459 . . . 4  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  =  ( A  \  x ) )
2 difss 3617 . . . 4  |-  ( A 
\  x )  C_  A
31, 2syl6eqss 3539 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  C_  A
)
4 selpw 4006 . . 3  |-  ( y  e.  ~P A  <->  y  C_  A )
53, 4sylibr 212 . 2  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  e.  ~P A )
61difeq2d 3608 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( A  \ 
y )  =  ( A  \  ( A 
\  x ) ) )
7 elpwi 4008 . . . . 5  |-  ( x  e.  ~P A  ->  x  C_  A )
87adantr 463 . . . 4  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  x  C_  A
)
9 dfss4 3729 . . . 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 2496 . 2  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  x  =  ( A  \  y ) )
125, 11jca 530 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 367    = wceq 1398    e. wcel 1823    \ cdif 3458    C_ wss 3461   ~Pcpw 3999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-pw 4001
This theorem is referenced by:  compsscnv  8742
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