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Theorem fpwwelem 8267
Description: Lemma for fpwwe 8268. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
fpwwe.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
fpwwe.2  |-  ( ph  ->  A  e.  _V )
Assertion
Ref Expression
fpwwelem  |-  ( ph  ->  ( X W R  <-> 
( ( X  C_  A  /\  R  C_  ( X  X.  X ) )  /\  ( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) ) ) )
Distinct variable groups:    x, r, A    y, r, F, x    ph, r, x, y    R, r, x, y    X, r, x, y    W, r, x, y
Allowed substitution hint:    A( y)

Proof of Theorem fpwwelem
StepHypRef Expression
1 fpwwe.1 . . . . 5  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
21relopabi 4811 . . . 4  |-  Rel  W
32a1i 10 . . 3  |-  ( ph  ->  Rel  W )
4 brrelex12 4726 . . 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  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  X  C_  A
)
7 fpwwe.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  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  A  e.  _V )
9 ssexg 4160 . . . 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  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  X  e.  _V )
11 simprlr 739 . . . 4  |-  ( (
ph  /\  ( ( X  C_  A  /\  R  C_  ( X  X.  X
) )  /\  ( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  R  C_  ( X  X.  X ) )
12 xpexg 4800 . . . . 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  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  ( X  X.  X )  e.  _V )
14 ssexg 4160 . . . 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  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  R  e.  _V )
1610, 15jca 518 . 2  |-  ( (
ph  /\  ( ( X  C_  A  /\  R  C_  ( X  X.  X
) )  /\  ( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  ( X  e. 
_V  /\  R  e.  _V ) )
17 simpl 443 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  x  =  X )
1817sseq1d 3205 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  C_  A  <->  X 
C_  A ) )
19 simpr 447 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  r  =  R )
2017, 17xpeq12d 4714 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
2119, 20sseq12d 3207 . . . . 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 4382 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
24 weeq1 4381 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2523, 24sylan9bb 680 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  We  x  <->  R  We  X ) )
2619cnveqd 4857 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  `' r  =  `' R )
2726imaeq1d 5011 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( `' r " { y } )  =  ( `' R " { y } ) )
2827fveq2d 5529 . . . . . . 7  |-  ( ( x  =  X  /\  r  =  R )  ->  ( F `  ( `' r " {
y } ) )  =  ( F `  ( `' R " { y } ) ) )
2928eqeq1d 2291 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( F `  ( `' r " {
y } ) )  =  y  <->  ( F `  ( `' R " { y } ) )  =  y ) )
3017, 29raleqbidv 2748 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( A. y  e.  x  ( F `  ( `' r " {
y } ) )  =  y  <->  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) )
3125, 30anbi12d 691 . . . 4  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y )  <-> 
( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) ) )
3222, 31anbi12d 691 . . 3  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) )  <->  ( ( X  C_  A  /\  R  C_  ( X  X.  X
) )  /\  ( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) ) ) )
3332, 1brabga 4279 . 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  ( F `  ( `' R " { y } ) )  =  y ) ) ) )
345, 16, 33pm5.21nd 868 1  |-  ( ph  ->  ( X W R  <-> 
( ( X  C_  A  /\  R  C_  ( X  X.  X ) )  /\  ( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   {csn 3640   class class class wbr 4023   {copab 4076    We wwe 4351    X. cxp 4687   `'ccnv 4688   "cima 4692   Rel wrel 4694   ` cfv 5255
This theorem is referenced by:  canth4  8269  canthnumlem  8270  canthp1lem2  8275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263
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