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Theorem dfoprab3s 6402
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 6121 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
2 nfsbc1v 3180 . . . . 5  |-  F/ x [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
3219.41 1900 . . . 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 6399 . . . . . . . 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 1592 . . . . . 6  |-  ( E. y ( w  = 
<. x ,  y >.  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph )  <->  E. y ( w  =  <. x ,  y
>.  /\  ph ) )
7 nfcv 2572 . . . . . . . 8  |-  F/_ y
( 1st `  w
)
8 nfsbc1v 3180 . . . . . . . 8  |-  F/ y
[. ( 2nd `  w
)  /  y ]. ph
97, 8nfsbc 3182 . . . . . . 7  |-  F/ y
[. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
10919.41 1900 . . . . . 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 1592 . . . 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 4936 . . . . 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 4272 . 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 2456 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 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956   [.wsbc 3161   <.cop 3817   {copab 4265    X. cxp 4876   ` cfv 5454   {coprab 6082   1stc1st 6347   2ndc2nd 6348
This theorem is referenced by:  dfoprab3  6403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-oprab 6085  df-1st 6349  df-2nd 6350
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