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Theorem fpwwelem 8453
Description: Lemma for fpwwe 8454. (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 4940 . . . 4  |-  Rel  W
32a1i 11 . . 3  |-  ( ph  ->  Rel  W )
4 brrelex12 4855 . . 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 fpwwe.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  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  A  e.  _V )
8 simprll 739 . . . 4  |-  ( (
ph  /\  ( ( X  C_  A  /\  R  C_  ( X  X.  X
) )  /\  ( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  X  C_  A
)
97, 8ssexd 4291 . . 3  |-  ( (
ph  /\  ( ( X  C_  A  /\  R  C_  ( X  X.  X
) )  /\  ( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  X  e.  _V )
10 xpexg 4929 . . . . 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  ( F `  ( `' R " { y } ) )  =  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  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  R  C_  ( X  X.  X ) )
1311, 12ssexd 4291 . . 3  |-  ( (
ph  /\  ( ( X  C_  A  /\  R  C_  ( X  X.  X
) )  /\  ( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) ) )  ->  R  e.  _V )
149, 13jca 519 . 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 ) )
15 simpl 444 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  x  =  X )
1615sseq1d 3318 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  C_  A  <->  X 
C_  A ) )
17 simpr 448 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  r  =  R )
1815, 15xpeq12d 4843 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( x  X.  x
)  =  ( X  X.  X ) )
1917, 18sseq12d 3320 . . . . 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 4512 . . . . . 6  |-  ( x  =  X  ->  (
r  We  x  <->  r  We  X ) )
22 weeq1 4511 . . . . . 6  |-  ( r  =  R  ->  (
r  We  X  <->  R  We  X ) )
2321, 22sylan9bb 681 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( r  We  x  <->  R  We  X ) )
2417cnveqd 4988 . . . . . . . . 9  |-  ( ( x  =  X  /\  r  =  R )  ->  `' r  =  `' R )
2524imaeq1d 5142 . . . . . . . 8  |-  ( ( x  =  X  /\  r  =  R )  ->  ( `' r " { y } )  =  ( `' R " { y } ) )
2625fveq2d 5672 . . . . . . 7  |-  ( ( x  =  X  /\  r  =  R )  ->  ( F `  ( `' r " {
y } ) )  =  ( F `  ( `' R " { y } ) ) )
2726eqeq1d 2395 . . . . . 6  |-  ( ( x  =  X  /\  r  =  R )  ->  ( ( F `  ( `' r " {
y } ) )  =  y  <->  ( F `  ( `' R " { y } ) )  =  y ) )
2815, 27raleqbidv 2859 . . . . 5  |-  ( ( x  =  X  /\  r  =  R )  ->  ( A. y  e.  x  ( F `  ( `' r " {
y } ) )  =  y  <->  A. y  e.  X  ( F `  ( `' R " { y } ) )  =  y ) )
2923, 28anbi12d 692 . . . 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 ) ) )
3020, 29anbi12d 692 . . 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 ) ) ) )
3130, 1brabga 4410 . 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 ) ) ) )
325, 14, 31pm5.21nd 869 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    C_ wss 3263   {csn 3757   class class class wbr 4153   {copab 4206    We wwe 4481    X. cxp 4816   `'ccnv 4817   "cima 4821   Rel wrel 4823   ` cfv 5394
This theorem is referenced by:  canth4  8455  canthnumlem  8456  canthp1lem2  8461
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-xp 4824  df-rel 4825  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fv 5402
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