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Theorem fpwwe2lem5 8256
Description: Lemma for fpwwe2 8265. (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 simpr1 961 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  X  C_  A )
2 fpwwe2.2 . . . . 5  |-  ( ph  ->  A  e.  _V )
32adantr 451 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  A  e.  _V )
4 ssexg 4160 . . . 4  |-  ( ( X  C_  A  /\  A  e.  _V )  ->  X  e.  _V )
51, 3, 4syl2anc 642 . . 3  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  X  e.  _V )
6 simpr2 962 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  R  C_  ( X  X.  X ) )
7 xpexg 4800 . . . . 5  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
85, 5, 7syl2anc 642 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X  X.  X
)  e.  _V )
9 ssexg 4160 . . . 4  |-  ( ( R  C_  ( X  X.  X )  /\  ( X  X.  X )  e. 
_V )  ->  R  e.  _V )
106, 8, 9syl2anc 642 . . 3  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  R  e.  _V )
115, 10jca 518 . 2  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X  e.  _V  /\  R  e.  _V )
)
12 sseq1 3199 . . . . . 6  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
13 xpeq12 4708 . . . . . . . 8  |-  ( ( x  =  X  /\  x  =  X )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1413anidms 626 . . . . . . 7  |-  ( x  =  X  ->  (
x  X.  x )  =  ( X  X.  X ) )
1514sseq2d 3206 . . . . . 6  |-  ( x  =  X  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( X  X.  X
) ) )
16 weeq2 4382 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
1712, 15, 163anbi123d 1252 . . . . 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 ) ) )
1817anbi2d 684 . . . 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 )
) ) )
19 oveq1 5865 . . . . 5  |-  ( x  =  X  ->  (
x F r )  =  ( X F r ) )
2019eleq1d 2349 . . . 4  |-  ( x  =  X  ->  (
( x F r )  e.  A  <->  ( X F r )  e.  A ) )
2118, 20imbi12d 311 . . 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 ) ) )
22 sseq1 3199 . . . . . 6  |-  ( r  =  R  ->  (
r  C_  ( X  X.  X )  <->  R  C_  ( X  X.  X ) ) )
23 weeq1 4381 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2422, 233anbi23d 1255 . . . . 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 ) ) )
2524anbi2d 684 . . . 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 )
) ) )
26 oveq2 5866 . . . . 5  |-  ( r  =  R  ->  ( X F r )  =  ( X F R ) )
2726eleq1d 2349 . . . 4  |-  ( r  =  R  ->  (
( X F r )  e.  A  <->  ( X F R )  e.  A
) )
2825, 27imbi12d 311 . . 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 ) ) )
29 fpwwe2.3 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x F r )  e.  A )
3021, 28, 29vtocl2g 2847 . 2  |-  ( ( X  e.  _V  /\  R  e.  _V )  ->  ( ( ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X )  /\  R  We  X ) )  -> 
( X F R )  e.  A ) )
3111, 30mpcom 32 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991    i^i cin 3151    C_ wss 3152   {csn 3640   {copab 4076    We wwe 4351    X. cxp 4687   `'ccnv 4688   "cima 4692  (class class class)co 5858
This theorem is referenced by:  fpwwe2lem13  8264
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  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-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-xp 4695  df-iota 5219  df-fv 5263  df-ov 5861
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