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Theorem fpwwe2lem2 8463
Description: Lemma for fpwwe2 8474. (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 4959 . . . 4  |-  Rel  W
32a1i 11 . . 3  |-  ( ph  ->  Rel  W )
4 brrelex12 4874 . . 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 4310 . . 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 4948 . . . . 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 4310 . . 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 3335 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  C_  A  <->  X 
C_  A ) )
17 simpr 448 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  r  =  R )
1815, 15xpeq12d 4862 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1917, 18sseq12d 3337 . . . . 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 4531 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
22 weeq1 4530 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2321, 22sylan9bb 681 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  We  x  <->  R  We  X ) )
2417ineq1d 3501 . . . . . . . . . 10  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  i^i  (
u  X.  u ) )  =  ( R  i^i  ( u  X.  u ) ) )
2524oveq2d 6056 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  ( u F ( r  i^i  ( u  X.  u ) ) )  =  ( u F ( R  i^i  ( u  X.  u
) ) ) )
2625eqeq1d 2412 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( u F ( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
2726sbcbidv 3175 . . . . . . 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 5007 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  `' r  =  `' R )
2928imaeq1d 5161 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( `' r " { y } )  =  ( `' R " { y } ) )
30 dfsbcq 3123 . . . . . . . 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 2876 . . . . 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 4429 . 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 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   [.wsbc 3121    i^i cin 3279    C_ wss 3280   {csn 3774   class class class wbr 4172   {copab 4225    We wwe 4500    X. cxp 4835   `'ccnv 4836   "cima 4840   Rel wrel 4842  (class class class)co 6040
This theorem is referenced by:  fpwwe2lem3  8464  fpwwe2lem6  8466  fpwwe2lem7  8467  fpwwe2lem9  8469  fpwwe2lem11  8471  fpwwe2lem12  8472  fpwwe2lem13  8473  fpwwe2  8474  canthwelem  8481  pwfseqlem4  8493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-ov 6043
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