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Theorem fpwwe2lem2 8811
Description: Lemma for fpwwe2 8822. (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 4977 . . . 4  |-  Rel  W
32a1i 11 . . 3  |-  ( ph  ->  Rel  W )
4 brrelex12 4888 . . 3  |-  ( ( Rel  W  /\  X W R )  ->  ( X  e.  _V  /\  R  e.  _V ) )
53, 4sylan 471 . 2  |-  ( (
ph  /\  X W R )  ->  ( X  e.  _V  /\  R  e.  _V ) )
6 fpwwe2.2 . . . . 5  |-  ( ph  ->  A  e.  _V )
76adantr 465 . . . 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 761 . . . 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 4451 . . 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 6519 . . . . 5  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
119, 9, 10syl2anc 661 . . . 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 762 . . . 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 4451 . . 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 532 . 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 457 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  x  =  X )
1615sseq1d 3395 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  C_  A  <->  X 
C_  A ) )
17 simpr 461 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  r  =  R )
1815, 15xpeq12d 4877 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1917, 18sseq12d 3397 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  C_  (
x  X.  x )  <-> 
R  C_  ( X  X.  X ) ) )
2016, 19anbi12d 710 . . . 4  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( X  C_  A  /\  R  C_  ( X  X.  X ) ) ) )
21 weeq2 4721 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
22 weeq1 4720 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2321, 22sylan9bb 699 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  We  x  <->  R  We  X ) )
2417ineq1d 3563 . . . . . . . . . 10  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  i^i  (
u  X.  u ) )  =  ( R  i^i  ( u  X.  u ) ) )
2524oveq2d 6119 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  ( u F ( r  i^i  ( u  X.  u ) ) )  =  ( u F ( R  i^i  ( u  X.  u
) ) ) )
2625eqeq1d 2451 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( u F ( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
2726sbcbidv 3257 . . . . . . 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 5027 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  `' r  =  `' R )
2928imaeq1d 5180 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( `' r " { y } )  =  ( `' R " { y } ) )
30 dfsbcq 3200 . . . . . . . 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 253 . . . . . 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 2943 . . . . 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 710 . . . 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 710 . . 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 4615 . 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 893 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727   _Vcvv 2984   [.wsbc 3198    i^i cin 3339    C_ wss 3340   {csn 3889   class class class wbr 4304   {copab 4361    We wwe 4690    X. cxp 4850   `'ccnv 4851   "cima 4855   Rel wrel 4857  (class class class)co 6103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-xp 4858  df-rel 4859  df-cnv 4860  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fv 5438  df-ov 6106
This theorem is referenced by:  fpwwe2lem3  8812  fpwwe2lem6  8814  fpwwe2lem7  8815  fpwwe2lem9  8817  fpwwe2lem11  8819  fpwwe2lem12  8820  fpwwe2lem13  8821  fpwwe2  8822  canthwelem  8829  pwfseqlem4  8841
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