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Theorem fpwwe2lem5 9008
Description: Lemma for fpwwe2 9017. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
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 ) ) }
fpwwe2.2  |-  ( ph  ->  A  e.  _V )
fpwwe2.3  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x F r )  e.  A )
Assertion
Ref Expression
fpwwe2lem5  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X F R )  e.  A )
Distinct variable groups:    y, u, r, x, F    X, r, u, x, y    ph, r, u, x, y    A, r, x    R, r, u, x, y    W, r, u, x, y
Allowed substitution hints:    A( y, u)

Proof of Theorem fpwwe2lem5
StepHypRef Expression
1 fpwwe2.2 . . . . 5  |-  ( ph  ->  A  e.  _V )
21adantr 465 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  A  e.  _V )
3 simpr1 1002 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  X  C_  A )
42, 3ssexd 4594 . . 3  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  X  e.  _V )
5 xpexg 6709 . . . . 5  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
64, 4, 5syl2anc 661 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X  X.  X
)  e.  _V )
7 simpr2 1003 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  R  C_  ( X  X.  X ) )
86, 7ssexd 4594 . . 3  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  R  e.  _V )
94, 8jca 532 . 2  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X  e.  _V  /\  R  e.  _V )
)
10 sseq1 3525 . . . . . 6  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
11 xpeq12 5018 . . . . . . . 8  |-  ( ( x  =  X  /\  x  =  X )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1211anidms 645 . . . . . . 7  |-  ( x  =  X  ->  (
x  X.  x )  =  ( X  X.  X ) )
1312sseq2d 3532 . . . . . 6  |-  ( x  =  X  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( X  X.  X
) ) )
14 weeq2 4868 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
1510, 13, 143anbi123d 1299 . . . . 5  |-  ( x  =  X  ->  (
( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  <->  ( X  C_  A  /\  r  C_  ( X  X.  X )  /\  r  We  X ) ) )
1615anbi2d 703 . . . 4  |-  ( x  =  X  ->  (
( ph  /\  (
x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  <->  ( ph  /\  ( X  C_  A  /\  r  C_  ( X  X.  X )  /\  r  We  X )
) ) )
17 oveq1 6289 . . . . 5  |-  ( x  =  X  ->  (
x F r )  =  ( X F r ) )
1817eleq1d 2536 . . . 4  |-  ( x  =  X  ->  (
( x F r )  e.  A  <->  ( X F r )  e.  A ) )
1916, 18imbi12d 320 . . 3  |-  ( x  =  X  ->  (
( ( ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  -> 
( x F r )  e.  A )  <-> 
( ( ph  /\  ( X  C_  A  /\  r  C_  ( X  X.  X )  /\  r  We  X ) )  -> 
( X F r )  e.  A ) ) )
20 sseq1 3525 . . . . . 6  |-  ( r  =  R  ->  (
r  C_  ( X  X.  X )  <->  R  C_  ( X  X.  X ) ) )
21 weeq1 4867 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2220, 213anbi23d 1302 . . . . 5  |-  ( r  =  R  ->  (
( X  C_  A  /\  r  C_  ( X  X.  X )  /\  r  We  X )  <->  ( X  C_  A  /\  R  C_  ( X  X.  X )  /\  R  We  X ) ) )
2322anbi2d 703 . . . 4  |-  ( r  =  R  ->  (
( ph  /\  ( X  C_  A  /\  r  C_  ( X  X.  X
)  /\  r  We  X ) )  <->  ( ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X )  /\  R  We  X )
) ) )
24 oveq2 6290 . . . . 5  |-  ( r  =  R  ->  ( X F r )  =  ( X F R ) )
2524eleq1d 2536 . . . 4  |-  ( r  =  R  ->  (
( X F r )  e.  A  <->  ( X F R )  e.  A
) )
2623, 25imbi12d 320 . . 3  |-  ( r  =  R  ->  (
( ( ph  /\  ( X  C_  A  /\  r  C_  ( X  X.  X )  /\  r  We  X ) )  -> 
( X F r )  e.  A )  <-> 
( ( ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X )  /\  R  We  X ) )  -> 
( X F R )  e.  A ) ) )
27 fpwwe2.3 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x F r )  e.  A )
2819, 26, 27vtocl2g 3175 . 2  |-  ( ( X  e.  _V  /\  R  e.  _V )  ->  ( ( ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X )  /\  R  We  X ) )  -> 
( X F R )  e.  A ) )
299, 28mpcom 36 1  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X F R )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   [.wsbc 3331    i^i cin 3475    C_ wss 3476   {csn 4027   {copab 4504    We wwe 4837    X. cxp 4997   `'ccnv 4998   "cima 5002  (class class class)co 6282
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  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-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-xp 5005  df-iota 5549  df-fv 5594  df-ov 6285
This theorem is referenced by:  fpwwe2lem13  9016
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