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Theorem fpwwe2lem2 8512
Description: Lemma for fpwwe2 8523. (Contributed by Mario Carneiro, 19-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 )
Assertion
Ref Expression
fpwwe2lem2  |-  ( ph  ->  ( X W 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 ) ) ) )
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 fpwwe2lem2
StepHypRef Expression
1 fpwwe2.1 . . . . 5  |-  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 ) ) }
21relopabi 5003 . . . 4  |-  Rel  W
32a1i 11 . . 3  |-  ( ph  ->  Rel  W )
4 brrelex12 4918 . . 3  |-  ( ( Rel  W  /\  X W R )  ->  ( X  e.  _V  /\  R  e.  _V ) )
53, 4sylan 459 . 2  |-  ( (
ph  /\  X W R )  ->  ( X  e.  _V  /\  R  e.  _V ) )
6 fpwwe2.2 . . . . 5  |-  ( ph  ->  A  e.  _V )
76adantr 453 . . . 4  |-  ( (
ph  /\  ( ( 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 ) ) )  ->  A  e.  _V )
8 simprll 740 . . . 4  |-  ( (
ph  /\  ( ( 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 ) ) )  ->  X  C_  A )
97, 8ssexd 4353 . . 3  |-  ( (
ph  /\  ( ( 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 ) ) )  ->  X  e.  _V )
10 xpexg 4992 . . . . 5  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
119, 9, 10syl2anc 644 . . . 4  |-  ( (
ph  /\  ( ( 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 ) ) )  ->  ( X  X.  X )  e. 
_V )
12 simprlr 741 . . . 4  |-  ( (
ph  /\  ( ( 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 ) ) )  ->  R  C_  ( X  X.  X
) )
1311, 12ssexd 4353 . . 3  |-  ( (
ph  /\  ( ( 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 ) ) )  ->  R  e.  _V )
149, 13jca 520 . 2  |-  ( (
ph  /\  ( ( 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 ) ) )  ->  ( X  e.  _V  /\  R  e.  _V ) )
15 simpl 445 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  x  =  X )
1615sseq1d 3377 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  C_  A  <->  X 
C_  A ) )
17 simpr 449 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  r  =  R )
1815, 15xpeq12d 4906 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1917, 18sseq12d 3379 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  C_  (
x  X.  x )  <-> 
R  C_  ( X  X.  X ) ) )
2016, 19anbi12d 693 . . . 4  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( X  C_  A  /\  R  C_  ( X  X.  X ) ) ) )
21 weeq2 4574 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
22 weeq1 4573 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2321, 22sylan9bb 682 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  We  x  <->  R  We  X ) )
2417ineq1d 3543 . . . . . . . . . 10  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  i^i  (
u  X.  u ) )  =  ( R  i^i  ( u  X.  u ) ) )
2524oveq2d 6100 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  ( u F ( r  i^i  ( u  X.  u ) ) )  =  ( u F ( R  i^i  ( u  X.  u
) ) ) )
2625eqeq1d 2446 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( u F ( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
2726sbcbidv 3217 . . . . . . 7  |-  ( ( x  =  X  /\  r  =  R )  ->  ( [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  [. ( `' r " { y } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
2817cnveqd 5051 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  `' r  =  `' R )
2928imaeq1d 5205 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( `' r " { y } )  =  ( `' R " { y } ) )
30 dfsbcq 3165 . . . . . . . 8  |-  ( ( `' r " {
y } )  =  ( `' R " { y } )  ->  ( [. ( `' r " {
y } )  /  u ]. ( u F ( R  i^i  (
u  X.  u ) ) )  =  y  <->  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
3129, 30syl 16 . . . . . . 7  |-  ( ( x  =  X  /\  r  =  R )  ->  ( [. ( `' r " { y } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  y  <->  [. ( `' R " { y } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
3227, 31bitrd 246 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  [. ( `' R " { y } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
3315, 32raleqbidv 2918 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  A. y  e.  X  [. ( `' R " { y } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
3423, 33anbi12d 693 . . . 4  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y )  <-> 
( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) ) )
3520, 34anbi12d 693 . . 3  |-  ( ( x  =  X  /\  r  =  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 ) )  <->  ( ( 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 ) ) ) )
3635, 1brabga 4472 . 2  |-  ( ( X  e.  _V  /\  R  e.  _V )  ->  ( X W 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 ) ) ) )
375, 14, 36pm5.21nd 870 1  |-  ( ph  ->  ( X W 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   [.wsbc 3163    i^i cin 3321    C_ wss 3322   {csn 3816   class class class wbr 4215   {copab 4268    We wwe 4543    X. cxp 4879   `'ccnv 4880   "cima 4884   Rel wrel 4886  (class class class)co 6084
This theorem is referenced by:  fpwwe2lem3  8513  fpwwe2lem6  8515  fpwwe2lem7  8516  fpwwe2lem9  8518  fpwwe2lem11  8520  fpwwe2lem12  8521  fpwwe2lem13  8522  fpwwe2  8523  canthwelem  8530  pwfseqlem4  8542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-ov 6087
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