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Theorem dfoprab3s 6867
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 6356 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
2 nfsbc1v 3275 . . . . 5  |-  F/ x [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
3219.41 2070 . . . 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 6864 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph  <->  ph ) )
54pm5.32i 649 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph )  <->  ( w  =  <. x ,  y
>.  /\  ph ) )
65exbii 1726 . . . . . 6  |-  ( E. y ( w  = 
<. x ,  y >.  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph )  <->  E. y ( w  =  <. x ,  y
>.  /\  ph ) )
7 nfcv 2612 . . . . . . . 8  |-  F/_ y
( 1st `  w
)
8 nfsbc1v 3275 . . . . . . . 8  |-  F/ y
[. ( 2nd `  w
)  /  y ]. ph
97, 8nfsbc 3277 . . . . . . 7  |-  F/ y
[. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
10919.41 2070 . . . . . 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 259 . . . . 5  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  ( E. y  w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) )
1211exbii 1726 . . . 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 4898 . . . . 5  |-  ( w  e.  ( _V  X.  _V )  <->  E. x E. y  w  =  <. x ,  y >. )
1413anbi1i 709 . . . 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 285 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  e.  ( _V  X.  _V )  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph ) )
1615opabbii 4460 . 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 2493 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031   [.wsbc 3255   <.cop 3965   {copab 4453    X. cxp 4837   ` cfv 5589   {coprab 6309   1stc1st 6810   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-oprab 6312  df-1st 6812  df-2nd 6813
This theorem is referenced by:  dfoprab3  6868
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