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Theorem fpwwe2lem5 9041
Description: Lemma for fpwwe2 9050. (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 463 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  A  e.  _V )
3 simpr1 1003 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  X  C_  A )
42, 3ssexd 4540 . . 3  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  X  e.  _V )
5 xpexg 6583 . . . . 5  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
64, 4, 5syl2anc 659 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X  X.  X
)  e.  _V )
7 simpr2 1004 . . . 4  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  R  C_  ( X  X.  X ) )
86, 7ssexd 4540 . . 3  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  ->  R  e.  _V )
94, 8jca 530 . 2  |-  ( (
ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X
)  /\  R  We  X ) )  -> 
( X  e.  _V  /\  R  e.  _V )
)
10 sseq1 3462 . . . . . 6  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
11 xpeq12 4841 . . . . . . . 8  |-  ( ( x  =  X  /\  x  =  X )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1211anidms 643 . . . . . . 7  |-  ( x  =  X  ->  (
x  X.  x )  =  ( X  X.  X ) )
1312sseq2d 3469 . . . . . 6  |-  ( x  =  X  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( X  X.  X
) ) )
14 weeq2 4811 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
1510, 13, 143anbi123d 1301 . . . . 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 702 . . . 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 6284 . . . . 5  |-  ( x  =  X  ->  (
x F r )  =  ( X F r ) )
1817eleq1d 2471 . . . 4  |-  ( x  =  X  ->  (
( x F r )  e.  A  <->  ( X F r )  e.  A ) )
1916, 18imbi12d 318 . . 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 3462 . . . . . 6  |-  ( r  =  R  ->  (
r  C_  ( X  X.  X )  <->  R  C_  ( X  X.  X ) ) )
21 weeq1 4810 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2220, 213anbi23d 1304 . . . . 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 702 . . . 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 6285 . . . . 5  |-  ( r  =  R  ->  ( X F r )  =  ( X F R ) )
2524eleq1d 2471 . . . 4  |-  ( r  =  R  ->  (
( X F r )  e.  A  <->  ( X F R )  e.  A
) )
2623, 25imbi12d 318 . . 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 3120 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   _Vcvv 3058   [.wsbc 3276    i^i cin 3412    C_ wss 3413   {csn 3971   {copab 4451    We wwe 4780    X. cxp 4820   `'ccnv 4821   "cima 4825  (class class class)co 6277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-iota 5532  df-fv 5576  df-ov 6280
This theorem is referenced by:  fpwwe2lem13  9049
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