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Theorem fpwwe2lem3 8464
Description: Lemma for fpwwe2 8474. (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 8463 . . . . . 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 202 . . . . 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 450 . . . 4  |-  ( ph  ->  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
76simprd 450 . . 3  |-  ( ph  ->  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y )
8 eqeq2 2413 . . . . . 6  |-  ( y  =  B  ->  (
( u F ( R  i^i  ( u  X.  u ) ) )  =  y  <->  ( u F ( R  i^i  ( u  X.  u
) ) )  =  B ) )
98sbcbidv 3175 . . . . 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 3785 . . . . . . 7  |-  ( y  =  B  ->  { y }  =  { B } )
1110imaeq2d 5162 . . . . . 6  |-  ( y  =  B  ->  ( `' R " { y } )  =  ( `' R " { B } ) )
12 dfsbcq 3123 . . . . . 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 245 . . . 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 3011 . . 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 458 . 2  |-  ( (
ph  /\  B  e.  X )  ->  [. ( `' R " { B } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  B )
17 cnvimass 5183 . . . . 5  |-  ( `' R " { B } )  C_  dom  R
182relopabi 4959 . . . . . . 7  |-  Rel  W
1918brrelex2i 4878 . . . . . 6  |-  ( X W R  ->  R  e.  _V )
20 dmexg 5089 . . . . . 6  |-  ( R  e.  _V  ->  dom  R  e.  _V )
211, 19, 203syl 19 . . . . 5  |-  ( ph  ->  dom  R  e.  _V )
22 ssexg 4309 . . . . 5  |-  ( ( ( `' R " { B } )  C_  dom  R  /\  dom  R  e.  _V )  ->  ( `' R " { B } )  e.  _V )
2317, 21, 22sylancr 645 . . . 4  |-  ( ph  ->  ( `' R " { B } )  e. 
_V )
24 id 20 . . . . . . 7  |-  ( u  =  ( `' R " { B } )  ->  u  =  ( `' R " { B } ) )
2524, 24xpeq12d 4862 . . . . . . . 8  |-  ( u  =  ( `' R " { B } )  ->  ( u  X.  u )  =  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) )
2625ineq2d 3502 . . . . . . 7  |-  ( u  =  ( `' R " { B } )  ->  ( R  i^i  ( u  X.  u
) )  =  ( R  i^i  ( ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )
2724, 26oveq12d 6058 . . . . . 6  |-  ( u  =  ( `' R " { B } )  ->  ( u F ( R  i^i  (
u  X.  u ) ) )  =  ( ( `' R " { B } ) F ( R  i^i  (
( `' R " { B } )  X.  ( `' R " { B } ) ) ) ) )
2827eqeq1d 2412 . . . . 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 3153 . . . 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 452 . 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 202 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   [.wsbc 3121    i^i cin 3279    C_ wss 3280   {csn 3774   class class class wbr 4172   {copab 4225    We wwe 4500    X. cxp 4835   `'ccnv 4836   dom cdm 4837   "cima 4840  (class class class)co 6040
This theorem is referenced by:  fpwwe2lem8  8468  fpwwe2lem12  8472  fpwwe2lem13  8473  fpwwe2  8474  canthwelem  8481  pwfseqlem4  8493
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-ov 6043
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