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Theorem fpwwe2lem3 8345
Description: Lemma for fpwwe2 8355. (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 8344 . . . . . 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 2367 . . . . . 6  |-  ( y  =  B  ->  (
( u F ( R  i^i  ( u  X.  u ) ) )  =  y  <->  ( u F ( R  i^i  ( u  X.  u
) ) )  =  B ) )
98sbcbidv 3121 . . . . 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 3727 . . . . . . 7  |-  ( y  =  B  ->  { y }  =  { B } )
1110imaeq2d 5094 . . . . . 6  |-  ( y  =  B  ->  ( `' R " { y } )  =  ( `' R " { B } ) )
12 dfsbcq 3069 . . . . . 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 2959 . . 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 5115 . . . . 5  |-  ( `' R " { B } )  C_  dom  R
182relopabi 4893 . . . . . . 7  |-  Rel  W
1918brrelex2i 4812 . . . . . 6  |-  ( X W R  ->  R  e.  _V )
20 dmexg 5021 . . . . . 6  |-  ( R  e.  _V  ->  dom  R  e.  _V )
211, 19, 203syl 18 . . . . 5  |-  ( ph  ->  dom  R  e.  _V )
22 ssexg 4241 . . . . 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 4796 . . . . . . . 8  |-  ( u  =  ( `' R " { B } )  ->  ( u  X.  u )  =  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) )
2625ineq2d 3446 . . . . . . 7  |-  ( u  =  ( `' R " { B } )  ->  ( R  i^i  ( u  X.  u
) )  =  ( R  i^i  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )
2724, 26oveq12d 5963 . . . . . 6  |-  ( u  =  ( `' R " { B } )  ->  ( u F ( R  i^i  (
u  X.  u ) ) )  =  ( ( `' R " { B } ) F ( R  i^i  (
( `' R " { B } )  X.  ( `' R " { B } ) ) ) ) )
2827eqeq1d 2366 . . . . 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 3099 . . . 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 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864   [.wsbc 3067    i^i cin 3227    C_ wss 3228   {csn 3716   class class class wbr 4104   {copab 4157    We wwe 4433    X. cxp 4769   `'ccnv 4770   dom cdm 4771   "cima 4774  (class class class)co 5945
This theorem is referenced by:  fpwwe2lem8  8349  fpwwe2lem12  8353  fpwwe2lem13  8354  fpwwe2  8355  canthwelem  8362  pwfseqlem4  8374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-xp 4777  df-rel 4778  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fv 5345  df-ov 5948
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