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Theorem fpwwe2lem2 8341
Description: Lemma for fpwwe2 8352. (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 4890 . . . 4  |-  Rel  W
32a1i 10 . . 3  |-  ( ph  ->  Rel  W )
4 brrelex12 4805 . . 3  |-  ( ( Rel  W  /\  X W R )  ->  ( X  e.  _V  /\  R  e.  _V ) )
53, 4sylan 457 . 2  |-  ( (
ph  /\  X W R )  ->  ( X  e.  _V  /\  R  e.  _V ) )
6 simprll 738 . . . 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 )
7 fpwwe2.2 . . . . 5  |-  ( ph  ->  A  e.  _V )
87adantr 451 . . . 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 )
9 ssexg 4239 . . . 4  |-  ( ( X  C_  A  /\  A  e.  _V )  ->  X  e.  _V )
106, 8, 9syl2anc 642 . . 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 )
11 simprlr 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 ) ) )  ->  R  C_  ( X  X.  X
) )
12 xpexg 4879 . . . . 5  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
1310, 10, 12syl2anc 642 . . . 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 )
14 ssexg 4239 . . . 4  |-  ( ( R  C_  ( X  X.  X )  /\  ( X  X.  X )  e. 
_V )  ->  R  e.  _V )
1511, 13, 14syl2anc 642 . . 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 )
1610, 15jca 518 . 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 ) )
17 simpl 443 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  x  =  X )
1817sseq1d 3281 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  C_  A  <->  X 
C_  A ) )
19 simpr 447 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  r  =  R )
2017, 17xpeq12d 4793 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
2119, 20sseq12d 3283 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  C_  (
x  X.  x )  <-> 
R  C_  ( X  X.  X ) ) )
2218, 21anbi12d 691 . . . 4  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( X  C_  A  /\  R  C_  ( X  X.  X ) ) ) )
23 weeq2 4461 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
24 weeq1 4460 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2523, 24sylan9bb 680 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  We  x  <->  R  We  X ) )
2619ineq1d 3445 . . . . . . . . . 10  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  i^i  (
u  X.  u ) )  =  ( R  i^i  ( u  X.  u ) ) )
2726oveq2d 5958 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  ( u F ( r  i^i  ( u  X.  u ) ) )  =  ( u F ( R  i^i  ( u  X.  u
) ) ) )
2827eqeq1d 2366 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( u F ( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
2928sbcbidv 3121 . . . . . . 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 ) )
3019cnveqd 4936 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  `' r  =  `' R )
3130imaeq1d 5090 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( `' r " { y } )  =  ( `' R " { y } ) )
32 dfsbcq 3069 . . . . . . . 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 ) )
3331, 32syl 15 . . . . . . 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 ) )
3429, 33bitrd 244 . . . . . 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 ) )
3517, 34raleqbidv 2824 . . . . 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 ) )
3625, 35anbi12d 691 . . . 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 ) ) )
3722, 36anbi12d 691 . . 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 ) ) ) )
3837, 1brabga 4358 . 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 ) ) ) )
395, 16, 38pm5.21nd 868 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 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864   [.wsbc 3067    i^i cin 3227    C_ wss 3228   {csn 3716   class class class wbr 4102   {copab 4155    We wwe 4430    X. cxp 4766   `'ccnv 4767   "cima 4771   Rel wrel 4773  (class class class)co 5942
This theorem is referenced by:  fpwwe2lem3  8342  fpwwe2lem6  8344  fpwwe2lem7  8345  fpwwe2lem9  8347  fpwwe2lem11  8349  fpwwe2lem12  8350  fpwwe2lem13  8351  fpwwe2  8352  canthwelem  8359  pwfseqlem4  8371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-xp 4774  df-rel 4775  df-cnv 4776  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fv 5342  df-ov 5945
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