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Theorem dfoprab3s 5960
Description: A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfoprab3s  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) }
Distinct variable groups:    ph, w    x, y, z, w
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem dfoprab3s
StepHypRef Expression
1 dfoprab2 5696 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
2 nfsbc1v 2858 . . . . 5  |-  F/ x [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
3219.41 1621 . . . 4  |-  ( E. x ( E. y  w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph )  <->  ( E. x E. y  w  = 
<. x ,  y >.  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph ) )
4 sbcopeq1a 5957 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph  <->  ph ) )
54pm5.32i 442 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph )  <->  ( w  =  <. x ,  y
>.  /\  ph ) )
65exbii 1541 . . . . . 6  |-  ( E. y ( w  = 
<. x ,  y >.  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph )  <->  E. y ( w  =  <. x ,  y
>.  /\  ph ) )
7 nfcv 2228 . . . . . . . 8  |-  F/_ y
( 1st `  w
)
8 nfsbc1v 2858 . . . . . . . 8  |-  F/ y
[. ( 2nd `  w
)  /  y ]. ph
97, 8nfsbc 2860 . . . . . . 7  |-  F/ y
[. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
10919.41 1621 . . . . . 6  |-  ( E. y ( w  = 
<. x ,  y >.  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph )  <->  ( E. y  w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) )
116, 10bitr3i 184 . . . . 5  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  ( E. y  w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) )
1211exbii 1541 . . . 4  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. x ( E. y  w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) )
13 elvv 4500 . . . . 5  |-  ( w  e.  ( _V  X.  _V )  <->  E. x E. y  w  =  <. x ,  y >. )
1413anbi1i 446 . . . 4  |-  ( ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph )  <->  ( E. x E. y  w  = 
<. x ,  y >.  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph ) )
153, 12, 143bitr4i 210 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  e.  ( _V  X.  _V )  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph ) )
1615opabbii 3905 . 2  |-  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }  =  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) }
171, 16eqtri 2108 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   [.wsbc 2840   <.cop 3449   {copab 3898    X. cxp 4436   ` cfv 5015   {coprab 5653   1stc1st 5909   2ndc2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-oprab 5656  df-1st 5911  df-2nd 5912
This theorem is referenced by:  dfoprab3  5961
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