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Theorem dfoprab3s 6191
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 5911 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
2 nfsbc1v 3023 . . . . 5  |-  F/ x [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
3219.41 1827 . . . 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 6188 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph  <->  ph ) )
54pm5.32i 618 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph )  <->  ( w  =  <. x ,  y
>.  /\  ph ) )
65exbii 1572 . . . . . 6  |-  ( E. y ( w  = 
<. x ,  y >.  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph )  <->  E. y ( w  =  <. x ,  y
>.  /\  ph ) )
7 nfcv 2432 . . . . . . . 8  |-  F/_ y
( 1st `  w
)
8 nfsbc1v 3023 . . . . . . . 8  |-  F/ y
[. ( 2nd `  w
)  /  y ]. ph
97, 8nfsbc 3025 . . . . . . 7  |-  F/ y
[. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph
10919.41 1827 . . . . . 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 242 . . . . 5  |-  ( E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  ( E. y  w  =  <. x ,  y >.  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) )
1211exbii 1572 . . . 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 4764 . . . . 5  |-  ( w  e.  ( _V  X.  _V )  <->  E. x E. y  w  =  <. x ,  y >. )
1413anbi1i 676 . . . 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 268 . . 3  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  e.  ( _V  X.  _V )  /\  [. ( 1st `  w
)  /  x ]. [. ( 2nd `  w
)  /  y ]. ph ) )
1615opabbii 4099 . 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 2316 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 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004   <.cop 3656   {copab 4092    X. cxp 4703   ` cfv 5271   {coprab 5875   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  dfoprab3  6192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-oprab 5878  df-1st 6138  df-2nd 6139
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