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Theorem fpwwe2lem5 8501
Description: Lemma for fpwwe2 8510. (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 452 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  A  e.  _V )
3 simpr1 963 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  X  C_  A )
42, 3ssexd 4342 . . 3  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  X  e.  _V )
5 xpexg 4981 . . . . 5  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
64, 4, 5syl2anc 643 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X  X.  X
)  e.  _V )
7 simpr2 964 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  R  C_  ( X  X.  X ) )
86, 7ssexd 4342 . . 3  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  R  e.  _V )
94, 8jca 519 . 2  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X  e.  _V  /\  R  e.  _V )
)
10 sseq1 3361 . . . . . 6  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
11 xpeq12 4889 . . . . . . . 8  |-  ( ( x  =  X  /\  x  =  X )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1211anidms 627 . . . . . . 7  |-  ( x  =  X  ->  (
x  X.  x )  =  ( X  X.  X ) )
1312sseq2d 3368 . . . . . 6  |-  ( x  =  X  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( X  X.  X
) ) )
14 weeq2 4563 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
1510, 13, 143anbi123d 1254 . . . . 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 685 . . . 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 6080 . . . . 5  |-  ( x  =  X  ->  (
x F r )  =  ( X F r ) )
1817eleq1d 2501 . . . 4  |-  ( x  =  X  ->  (
( x F r )  e.  A  <->  ( X F r )  e.  A ) )
1916, 18imbi12d 312 . . 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 3361 . . . . . 6  |-  ( r  =  R  ->  (
r  C_  ( X  X.  X )  <->  R  C_  ( X  X.  X ) ) )
21 weeq1 4562 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2220, 213anbi23d 1257 . . . . 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 685 . . . 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 6081 . . . . 5  |-  ( r  =  R  ->  ( X F r )  =  ( X F R ) )
2524eleq1d 2501 . . . 4  |-  ( r  =  R  ->  (
( X F r )  e.  A  <->  ( X F R )  e.  A
) )
2623, 25imbi12d 312 . . 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 3007 . 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 34 1  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X F R )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   [.wsbc 3153    i^i cin 3311    C_ wss 3312   {csn 3806   {copab 4257    We wwe 4532    X. cxp 4868   `'ccnv 4869   "cima 4873  (class class class)co 6073
This theorem is referenced by:  fpwwe2lem13  8509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-xp 4876  df-iota 5410  df-fv 5454  df-ov 6076
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