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Theorem fpwwe2lem3 8796
Description: Lemma for fpwwe2 8806. (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 8795 . . . . . 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 460 . . . 4  |-  ( ph  ->  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
76simprd 460 . . 3  |-  ( ph  ->  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y )
8 eqeq2 2450 . . . . . 6  |-  ( y  =  B  ->  (
( u F ( R  i^i  ( u  X.  u ) ) )  =  y  <->  ( u F ( R  i^i  ( u  X.  u
) ) )  =  B ) )
98sbcbidv 3242 . . . . 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 3884 . . . . . . 7  |-  ( y  =  B  ->  { y }  =  { B } )
1110imaeq2d 5166 . . . . . 6  |-  ( y  =  B  ->  ( `' R " { y } )  =  ( `' R " { B } ) )
12 dfsbcq 3185 . . . . . 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 16 . . . . 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 253 . . . 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 3069 . . 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 468 . 2  |-  ( (
ph  /\  B  e.  X )  ->  [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B )
17 cnvimass 5186 . . . . 5  |-  ( `' R " { B } )  C_  dom  R
182relopabi 4961 . . . . . . 7  |-  Rel  W
1918brrelex2i 4876 . . . . . 6  |-  ( X W R  ->  R  e.  _V )
20 dmexg 6508 . . . . . 6  |-  ( R  e.  _V  ->  dom  R  e.  _V )
211, 19, 203syl 20 . . . . 5  |-  ( ph  ->  dom  R  e.  _V )
22 ssexg 4435 . . . . 5  |-  ( ( ( `' R " { B } )  C_  dom  R  /\  dom  R  e.  _V )  ->  ( `' R " { B } )  e.  _V )
2317, 21, 22sylancr 658 . . . 4  |-  ( ph  ->  ( `' R " { B } )  e. 
_V )
24 id 22 . . . . . . 7  |-  ( u  =  ( `' R " { B } )  ->  u  =  ( `' R " { B } ) )
2524, 24xpeq12d 4861 . . . . . . . 8  |-  ( u  =  ( `' R " { B } )  ->  ( u  X.  u )  =  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) )
2625ineq2d 3549 . . . . . . 7  |-  ( u  =  ( `' R " { B } )  ->  ( R  i^i  ( u  X.  u
) )  =  ( R  i^i  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )
2724, 26oveq12d 6108 . . . . . 6  |-  ( u  =  ( `' R " { B } )  ->  ( u F ( R  i^i  (
u  X.  u ) ) )  =  ( ( `' R " { B } ) F ( R  i^i  (
( `' R " { B } )  X.  ( `' R " { B } ) ) ) ) )
2827eqeq1d 2449 . . . . 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 3216 . . . 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 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 ) )
3130adantr 462 . 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 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 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970   [.wsbc 3183    i^i cin 3324    C_ wss 3325   {csn 3874   class class class wbr 4289   {copab 4346    We wwe 4674    X. cxp 4834   `'ccnv 4835   dom cdm 4836   "cima 4839  (class class class)co 6090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-xp 4842  df-rel 4843  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fv 5423  df-ov 6093
This theorem is referenced by:  fpwwe2lem8  8800  fpwwe2lem12  8804  fpwwe2lem13  8805  fpwwe2  8806  canthwelem  8813  pwfseqlem4  8825
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