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Theorem dfoprab3 6057
Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
Hypothesis
Ref Expression
dfoprab3.1  |-  ( w  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
dfoprab3  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ps }
Distinct variable groups:    x, y, ph    ps, w    x, z, w, y
Allowed substitution hints:    ph( z, w)    ps( x, y, z)

Proof of Theorem dfoprab3
StepHypRef Expression
1 dfoprab3s 6056 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps ) }
2 vex 2663 . . . . . 6  |-  w  e. 
_V
3 1stexg 6033 . . . . . 6  |-  ( w  e.  _V  ->  ( 1st `  w )  e. 
_V )
42, 3ax-mp 5 . . . . 5  |-  ( 1st `  w )  e.  _V
5 2ndexg 6034 . . . . . 6  |-  ( w  e.  _V  ->  ( 2nd `  w )  e. 
_V )
62, 5ax-mp 5 . . . . 5  |-  ( 2nd `  w )  e.  _V
7 eqcom 2119 . . . . . . . . . 10  |-  ( x  =  ( 1st `  w
)  <->  ( 1st `  w
)  =  x )
8 eqcom 2119 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  w
)  <->  ( 2nd `  w
)  =  y )
97, 8anbi12i 455 . . . . . . . . 9  |-  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  <->  ( ( 1st `  w )  =  x  /\  ( 2nd `  w )  =  y ) )
10 eqopi 6038 . . . . . . . . 9  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
( 1st `  w
)  =  x  /\  ( 2nd `  w )  =  y ) )  ->  w  =  <. x ,  y >. )
119, 10sylan2b 285 . . . . . . . 8  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  ->  w  =  <. x ,  y >. )
12 dfoprab3.1 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
1311, 12syl 14 . . . . . . 7  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  -> 
( ph  <->  ps ) )
1413bicomd 140 . . . . . 6  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  -> 
( ps  <->  ph ) )
1514ex 114 . . . . 5  |-  ( w  e.  ( _V  X.  _V )  ->  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  ->  ( ps 
<-> 
ph ) ) )
164, 6, 15sbc2iedv 2953 . . . 4  |-  ( w  e.  ( _V  X.  _V )  ->  ( [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  / 
y ]. ps  <->  ph ) )
1716pm5.32i 449 . . 3  |-  ( ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps )  <->  ( w  e.  ( _V  X.  _V )  /\  ph ) )
1817opabbii 3965 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps ) }  =  { <. w ,  z
>.  |  ( w  e.  ( _V  X.  _V )  /\  ph ) }
191, 18eqtr2i 2139 1  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660   [.wsbc 2882   <.cop 3500   {copab 3958    X. cxp 4507   ` cfv 5093   {coprab 5743   1stc1st 6004   2ndc2nd 6005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-oprab 5746  df-1st 6006  df-2nd 6007
This theorem is referenced by:  dfoprab4  6058  df1st2  6084  df2nd2  6085
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