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Theorem dfoprab3s 6361
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 6080 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
2 nfsbc1v 3140 . . . . 5  |-  F/ x [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
3219.41 1896 . . . 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 6358 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph  <->  ph ) )
54pm5.32i 619 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph )  <->  ( w  =  <. x ,  y
>.  /\  ph ) )
65exbii 1589 . . . . . 6  |-  ( E. y ( w  = 
<. x ,  y >.  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph )  <->  E. y ( w  =  <. x ,  y
>.  /\  ph ) )
7 nfcv 2540 . . . . . . . 8  |-  F/_ y
( 1st `  w
)
8 nfsbc1v 3140 . . . . . . . 8  |-  F/ y
[. ( 2nd `  w
)  /  y ]. ph
97, 8nfsbc 3142 . . . . . . 7  |-  F/ y
[. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
10919.41 1896 . . . . . 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 243 . . . . 5  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  ( E. y  w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) )
1211exbii 1589 . . . 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 4895 . . . . 5  |-  ( w  e.  ( _V  X.  _V )  <->  E. x E. y  w  =  <. x ,  y >. )
1413anbi1i 677 . . . 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 269 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  e.  ( _V  X.  _V )  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph ) )
1615opabbii 4232 . 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 2424 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 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916   [.wsbc 3121   <.cop 3777   {copab 4225    X. cxp 4835   ` cfv 5413   {coprab 6041   1stc1st 6306   2ndc2nd 6307
This theorem is referenced by:  dfoprab3  6362
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-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-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fv 5421  df-oprab 6044  df-1st 6308  df-2nd 6309
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