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Theorem fpwwe2lem1 8790
Description: Lemma for fpwwe2 8802. (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 753 . . . . 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 selpw 3862 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
31, 2sylibr 212 . . . 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 )
4 simplr 754 . . . . . 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
) )
5 xpss12 4940 . . . . . . 7  |-  ( ( x  C_  A  /\  x  C_  A )  -> 
( x  X.  x
)  C_  ( A  X.  A ) )
61, 1, 5syl2anc 661 . . . . . 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 ) )
74, 6sstrd 3361 . . . . 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
) )
8 selpw 3862 . . . . 5  |-  ( r  e.  ~P ( A  X.  A )  <->  r  C_  ( A  X.  A
) )
97, 8sylibr 212 . . . 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
) )
103, 9jca 532 . . 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 ) ) )
1110ssopab2i 4611 . 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
) ) }
12 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 ) ) }
13 df-xp 4841 . 2  |-  ( ~P A  X.  ~P ( A  X.  A ) )  =  { <. x ,  r >.  |  ( x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) }
1411, 12, 133sstr4i 3390 1  |-  W  C_  ( ~P A  X.  ~P ( A  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   [.wsbc 3181    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   {csn 3872   {copab 4344    We wwe 4673    X. cxp 4833   `'ccnv 4834   "cima 4838  (class class class)co 6086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969  df-in 3330  df-ss 3337  df-pw 3857  df-opab 4346  df-xp 4841
This theorem is referenced by: (None)
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