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Theorem fpwwe2lem3 8255
Description: Lemma for fpwwe2 8265. (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 8254 . . . . . 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 201 . . . . 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 449 . . . 4  |-  ( ph  ->  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
76simprd 449 . . 3  |-  ( ph  ->  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y )
8 eqeq2 2292 . . . . . 6  |-  ( y  =  B  ->  (
( u F ( R  i^i  ( u  X.  u ) ) )  =  y  <->  ( u F ( R  i^i  ( u  X.  u
) ) )  =  B ) )
98sbcbidv 3045 . . . . 5  |-  ( y  =  B  ->  ( [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y  <->  [. ( `' R " { y } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B ) )
10 sneq 3651 . . . . . . 7  |-  ( y  =  B  ->  { y }  =  { B } )
1110imaeq2d 5012 . . . . . 6  |-  ( y  =  B  ->  ( `' R " { y } )  =  ( `' R " { B } ) )
12 dfsbcq 2993 . . . . . 6  |-  ( ( `' R " { y } )  =  ( `' R " { B } )  ->  ( [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  B  <->  [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B ) )
1311, 12syl 15 . . . . 5  |-  ( y  =  B  ->  ( [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  B  <->  [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B ) )
149, 13bitrd 244 . . . 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 ) )
1514rspccva 2883 . . 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 )
167, 15sylan 457 . 2  |-  ( (
ph  /\  B  e.  X )  ->  [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B )
17 cnvimass 5033 . . . . 5  |-  ( `' R " { B } )  C_  dom  R
182relopabi 4811 . . . . . . 7  |-  Rel  W
1918brrelex2i 4730 . . . . . 6  |-  ( X W R  ->  R  e.  _V )
20 dmexg 4939 . . . . . 6  |-  ( R  e.  _V  ->  dom  R  e.  _V )
211, 19, 203syl 18 . . . . 5  |-  ( ph  ->  dom  R  e.  _V )
22 ssexg 4160 . . . . 5  |-  ( ( ( `' R " { B } )  C_  dom  R  /\  dom  R  e.  _V )  ->  ( `' R " { B } )  e.  _V )
2317, 21, 22sylancr 644 . . . 4  |-  ( ph  ->  ( `' R " { B } )  e. 
_V )
24 id 19 . . . . . . 7  |-  ( u  =  ( `' R " { B } )  ->  u  =  ( `' R " { B } ) )
2524, 24xpeq12d 4714 . . . . . . . 8  |-  ( u  =  ( `' R " { B } )  ->  ( u  X.  u )  =  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) )
2625ineq2d 3370 . . . . . . 7  |-  ( u  =  ( `' R " { B } )  ->  ( R  i^i  ( u  X.  u
) )  =  ( R  i^i  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )
2724, 26oveq12d 5876 . . . . . 6  |-  ( u  =  ( `' R " { B } )  ->  ( u F ( R  i^i  (
u  X.  u ) ) )  =  ( ( `' R " { B } ) F ( R  i^i  (
( `' R " { B } )  X.  ( `' R " { B } ) ) ) ) )
2827eqeq1d 2291 . . . . 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 ) )
2928sbcieg 3023 . . . 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 ) )
3023, 29syl 15 . . 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 ) )
3130adantr 451 . 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 ) )
3216, 31mpbid 201 1  |-  ( (
ph  /\  B  e.  X )  ->  (
( `' R " { B } ) F ( R  i^i  (
( `' R " { B } )  X.  ( `' R " { B } ) ) ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991    i^i cin 3151    C_ wss 3152   {csn 3640   class class class wbr 4023   {copab 4076    We wwe 4351    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692  (class class class)co 5858
This theorem is referenced by:  fpwwe2lem8  8259  fpwwe2lem12  8263  fpwwe2lem13  8264  fpwwe2  8265  canthwelem  8272  pwfseqlem4  8284
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-sbc 2992  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  df-ov 5861
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