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Theorem fpwwe2lem5 8272
Description: Lemma for fpwwe2 8281. (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 4176 . . . 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 4816 . . . . 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 4176 . . . 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 3212 . . . . . 6  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
13 xpeq12 4724 . . . . . . . 8  |-  ( ( x  =  X  /\  x  =  X )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1413anidms 626 . . . . . . 7  |-  ( x  =  X  ->  (
x  X.  x )  =  ( X  X.  X ) )
1514sseq2d 3219 . . . . . 6  |-  ( x  =  X  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( X  X.  X
) ) )
16 weeq2 4398 . . . . . 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 5881 . . . . 5  |-  ( x  =  X  ->  (
x F r )  =  ( X F r ) )
2019eleq1d 2362 . . . 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 3212 . . . . . 6  |-  ( r  =  R  ->  (
r  C_  ( X  X.  X )  <->  R  C_  ( X  X.  X ) ) )
23 weeq1 4397 . . . . . 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 5882 . . . . 5  |-  ( r  =  R  ->  ( X F r )  =  ( X F R ) )
2726eleq1d 2362 . . . 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 2860 . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   [.wsbc 3004    i^i cin 3164    C_ wss 3165   {csn 3653   {copab 4092    We wwe 4367    X. cxp 4703   `'ccnv 4704   "cima 4708  (class class class)co 5874
This theorem is referenced by:  fpwwe2lem13  8280
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-xp 4711  df-iota 5235  df-fv 5279  df-ov 5877
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