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Theorem fpwwe2lem3 9023
Description: Lemma for fpwwe2 9033. (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 )
fpwwe2lem4.4  |-  ( ph  ->  X W R )
Assertion
Ref Expression
fpwwe2lem3  |-  ( (
ph  /\  B  e.  X )  ->  (
( `' R " { B } ) F ( R  i^i  (
( `' R " { B } )  X.  ( `' R " { B } ) ) ) )  =  B )
Distinct variable groups:    y, u, B    u, r, x, y, 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)    B( x, r)

Proof of Theorem fpwwe2lem3
StepHypRef Expression
1 fpwwe2lem4.4 . . . . . 6  |-  ( ph  ->  X W R )
2 fpwwe2.1 . . . . . . 7  |-  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 ) ) }
3 fpwwe2.2 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
42, 3fpwwe2lem2 9022 . . . . . 6  |-  ( 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 ) ) ) )
51, 4mpbid 210 . . . . 5  |-  ( 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 ) ) )
65simprd 463 . . . 4  |-  ( ph  ->  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
76simprd 463 . . 3  |-  ( ph  ->  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y )
8 sneq 4043 . . . . . 6  |-  ( y  =  B  ->  { y }  =  { B } )
98imaeq2d 5343 . . . . 5  |-  ( y  =  B  ->  ( `' R " { y } )  =  ( `' R " { B } ) )
10 eqeq2 2482 . . . . 5  |-  ( y  =  B  ->  (
( u F ( R  i^i  ( u  X.  u ) ) )  =  y  <->  ( u F ( R  i^i  ( u  X.  u
) ) )  =  B ) )
119, 10sbceqbid 3343 . . . 4  |-  ( y  =  B  ->  ( [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y  <->  [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B ) )
1211rspccva 3218 . . 3  |-  ( ( A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y  /\  B  e.  X )  ->  [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B )
137, 12sylan 471 . 2  |-  ( (
ph  /\  B  e.  X )  ->  [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B )
14 cnvimass 5363 . . . . 5  |-  ( `' R " { B } )  C_  dom  R
152relopabi 5134 . . . . . . 7  |-  Rel  W
1615brrelex2i 5047 . . . . . 6  |-  ( X W R  ->  R  e.  _V )
17 dmexg 6726 . . . . . 6  |-  ( R  e.  _V  ->  dom  R  e.  _V )
181, 16, 173syl 20 . . . . 5  |-  ( ph  ->  dom  R  e.  _V )
19 ssexg 4599 . . . . 5  |-  ( ( ( `' R " { B } )  C_  dom  R  /\  dom  R  e.  _V )  ->  ( `' R " { B } )  e.  _V )
2014, 18, 19sylancr 663 . . . 4  |-  ( ph  ->  ( `' R " { B } )  e. 
_V )
21 id 22 . . . . . . 7  |-  ( u  =  ( `' R " { B } )  ->  u  =  ( `' R " { B } ) )
2221sqxpeqd 5031 . . . . . . . 8  |-  ( u  =  ( `' R " { B } )  ->  ( u  X.  u )  =  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) )
2322ineq2d 3705 . . . . . . 7  |-  ( u  =  ( `' R " { B } )  ->  ( R  i^i  ( u  X.  u
) )  =  ( R  i^i  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )
2421, 23oveq12d 6313 . . . . . 6  |-  ( u  =  ( `' R " { B } )  ->  ( u F ( R  i^i  (
u  X.  u ) ) )  =  ( ( `' R " { B } ) F ( R  i^i  (
( `' R " { B } )  X.  ( `' R " { B } ) ) ) ) )
2524eqeq1d 2469 . . . . 5  |-  ( u  =  ( `' R " { B } )  ->  ( ( u F ( R  i^i  ( u  X.  u
) ) )  =  B  <->  ( ( `' R " { B } ) F ( R  i^i  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )  =  B ) )
2625sbcieg 3369 . . . 4  |-  ( ( `' R " { B } )  e.  _V  ->  ( [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B  <->  ( ( `' R " { B } ) F ( R  i^i  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )  =  B ) )
2720, 26syl 16 . . 3  |-  ( ph  ->  ( [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B  <->  ( ( `' R " { B } ) F ( R  i^i  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )  =  B ) )
2827adantr 465 . 2  |-  ( (
ph  /\  B  e.  X )  ->  ( [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  (
u  X.  u ) ) )  =  B  <-> 
( ( `' R " { B } ) F ( R  i^i  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )  =  B ) )
2913, 28mpbid 210 1  |-  ( (
ph  /\  B  e.  X )  ->  (
( `' R " { B } ) F ( R  i^i  (
( `' R " { B } )  X.  ( `' R " { B } ) ) ) )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   [.wsbc 3336    i^i cin 3480    C_ wss 3481   {csn 4033   class class class wbr 4453   {copab 4510    We wwe 4843    X. cxp 5003   `'ccnv 5004   dom cdm 5005   "cima 5008  (class class class)co 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-xp 5011  df-rel 5012  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fv 5602  df-ov 6298
This theorem is referenced by:  fpwwe2lem8  9027  fpwwe2lem12  9031  fpwwe2lem13  9032  fpwwe2  9033  canthwelem  9040  pwfseqlem4  9052
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