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Theorem fpwwe2lem1 8269
Description: Lemma for fpwwe2 8281. (Contributed by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
fpwwe2.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
Assertion
Ref Expression
fpwwe2lem1  |-  W  C_  ( ~P A  X.  ~P ( A  X.  A
) )
Distinct variable groups:    y, u, r, x, F    A, r, x    W, r, u, x, y
Allowed substitution hints:    A( y, u)

Proof of Theorem fpwwe2lem1
StepHypRef Expression
1 simpll 730 . . . . 5  |-  ( ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) )  ->  x  C_  A
)
2 vex 2804 . . . . . 6  |-  x  e. 
_V
32elpw 3644 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
41, 3sylibr 203 . . . 4  |-  ( ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) )  ->  x  e.  ~P A )
5 simplr 731 . . . . . 6  |-  ( ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) )  ->  r  C_  ( x  X.  x
) )
6 xpss12 4808 . . . . . . 7  |-  ( ( x  C_  A  /\  x  C_  A )  -> 
( x  X.  x
)  C_  ( A  X.  A ) )
71, 1, 6syl2anc 642 . . . . . 6  |-  ( ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) )  ->  ( x  X.  x )  C_  ( A  X.  A ) )
85, 7sstrd 3202 . . . . 5  |-  ( ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) )  ->  r  C_  ( A  X.  A
) )
9 vex 2804 . . . . . 6  |-  r  e. 
_V
109elpw 3644 . . . . 5  |-  ( r  e.  ~P ( A  X.  A )  <->  r  C_  ( A  X.  A
) )
118, 10sylibr 203 . . . 4  |-  ( ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) )  ->  r  e.  ~P ( A  X.  A
) )
124, 11jca 518 . . 3  |-  ( ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) )  ->  ( x  e.  ~P A  /\  r  e.  ~P ( A  X.  A ) ) )
1312ssopab2i 4308 . 2  |-  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }  C_  { <. x ,  r >.  |  ( x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) }
14 fpwwe2.1 . 2  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
15 df-xp 4711 . 2  |-  ( ~P A  X.  ~P ( A  X.  A ) )  =  { <. x ,  r >.  |  ( x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) }
1613, 14, 153sstr4i 3230 1  |-  W  C_  ( ~P A  X.  ~P ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   [.wsbc 3004    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   {csn 3653   {copab 4092    We wwe 4367    X. cxp 4703   `'ccnv 4704   "cima 4708  (class class class)co 5874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-opab 4094  df-xp 4711
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