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Theorem compsscnvlem 7996
Description: Lemma for compsscnv 7997. (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 difss 3303 . . . 4  |-  ( A 
\  x )  C_  A
2 simpr 447 . . . . 5  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  =  ( A  \  x ) )
32sseq1d 3205 . . . 4  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( y  C_  A 
<->  ( A  \  x
)  C_  A )
)
41, 3mpbiri 224 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  C_  A
)
5 vex 2791 . . . 4  |-  y  e. 
_V
65elpw 3631 . . 3  |-  ( y  e.  ~P A  <->  y  C_  A )
74, 6sylibr 203 . 2  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  y  e.  ~P A )
82difeq2d 3294 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( A  \ 
y )  =  ( A  \  ( A 
\  x ) ) )
9 elpwi 3633 . . . . 5  |-  ( x  e.  ~P A  ->  x  C_  A )
109adantr 451 . . . 4  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  x  C_  A
)
11 dfss4 3403 . . . 4  |-  ( x 
C_  A  <->  ( A  \  ( A  \  x
) )  =  x )
1210, 11sylib 188 . . 3  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( A  \ 
( A  \  x
) )  =  x )
138, 12eqtr2d 2316 . 2  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  x  =  ( A  \  y ) )
147, 13jca 518 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 358    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   ~Pcpw 3625
This theorem is referenced by:  compsscnv  7997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-pw 3627
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