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Theorem fpwwe2lem2 8499
Description: Lemma for fpwwe2 8510. (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 4992 . . . 4  |-  Rel  W
32a1i 11 . . 3  |-  ( ph  ->  Rel  W )
4 brrelex12 4907 . . 3  |-  ( ( Rel  W  /\  X W R )  ->  ( X  e.  _V  /\  R  e.  _V ) )
53, 4sylan 458 . 2  |-  ( (
ph  /\  X W R )  ->  ( X  e.  _V  /\  R  e.  _V ) )
6 fpwwe2.2 . . . . 5  |-  ( ph  ->  A  e.  _V )
76adantr 452 . . . 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 739 . . . 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 4342 . . 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 4981 . . . . 5  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
119, 9, 10syl2anc 643 . . . 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 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 ) ) )  ->  R  C_  ( X  X.  X
) )
1311, 12ssexd 4342 . . 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 519 . 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 444 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  x  =  X )
1615sseq1d 3367 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  C_  A  <->  X 
C_  A ) )
17 simpr 448 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  r  =  R )
1815, 15xpeq12d 4895 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1917, 18sseq12d 3369 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  C_  (
x  X.  x )  <-> 
R  C_  ( X  X.  X ) ) )
2016, 19anbi12d 692 . . . 4  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( X  C_  A  /\  R  C_  ( X  X.  X ) ) ) )
21 weeq2 4563 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
22 weeq1 4562 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2321, 22sylan9bb 681 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  We  x  <->  R  We  X ) )
2417ineq1d 3533 . . . . . . . . . 10  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  i^i  (
u  X.  u ) )  =  ( R  i^i  ( u  X.  u ) ) )
2524oveq2d 6089 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  ( u F ( r  i^i  ( u  X.  u ) ) )  =  ( u F ( R  i^i  ( u  X.  u
) ) ) )
2625eqeq1d 2443 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( u F ( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
2726sbcbidv 3207 . . . . . . 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 5040 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  `' r  =  `' R )
2928imaeq1d 5194 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( `' r " { y } )  =  ( `' R " { y } ) )
30 dfsbcq 3155 . . . . . . . 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 245 . . . . . 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 2908 . . . . 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 692 . . . 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 692 . . 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 4461 . 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 869 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   [.wsbc 3153    i^i cin 3311    C_ wss 3312   {csn 3806   class class class wbr 4204   {copab 4257    We wwe 4532    X. cxp 4868   `'ccnv 4869   "cima 4873   Rel wrel 4875  (class class class)co 6073
This theorem is referenced by:  fpwwe2lem3  8500  fpwwe2lem6  8502  fpwwe2lem7  8503  fpwwe2lem9  8505  fpwwe2lem11  8507  fpwwe2lem12  8508  fpwwe2lem13  8509  fpwwe2  8510  canthwelem  8517  pwfseqlem4  8529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-ov 6076
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