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Theorem dfoprab3 6176
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 6175 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps ) }
2 fvex 5539 . . . . 5  |-  ( 1st `  w )  e.  _V
3 fvex 5539 . . . . 5  |-  ( 2nd `  w )  e.  _V
4 eqcom 2285 . . . . . . . . . 10  |-  ( x  =  ( 1st `  w
)  <->  ( 1st `  w
)  =  x )
5 eqcom 2285 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  w
)  <->  ( 2nd `  w
)  =  y )
64, 5anbi12i 678 . . . . . . . . 9  |-  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  <->  ( ( 1st `  w )  =  x  /\  ( 2nd `  w )  =  y ) )
7 eqopi 6156 . . . . . . . . 9  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
( 1st `  w
)  =  x  /\  ( 2nd `  w )  =  y ) )  ->  w  =  <. x ,  y >. )
86, 7sylan2b 461 . . . . . . . 8  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  ->  w  =  <. x ,  y >. )
9 dfoprab3.1 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
108, 9syl 15 . . . . . . 7  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  -> 
( ph  <->  ps ) )
1110bicomd 192 . . . . . 6  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  -> 
( ps  <->  ph ) )
1211ex 423 . . . . 5  |-  ( w  e.  ( _V  X.  _V )  ->  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  ->  ( ps 
<-> 
ph ) ) )
132, 3, 12sbc2iedv 3059 . . . 4  |-  ( w  e.  ( _V  X.  _V )  ->  ( [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  / 
y ]. ps  <->  ph ) )
1413pm5.32i 618 . . 3  |-  ( ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps )  <->  ( w  e.  ( _V  X.  _V )  /\  ph ) )
1514opabbii 4083 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps ) }  =  { <. w ,  z
>.  |  ( w  e.  ( _V  X.  _V )  /\  ph ) }
161, 15eqtr2i 2304 1  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991   <.cop 3643   {copab 4076    X. cxp 4687   ` cfv 5255   {coprab 5859   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  dfoprab4  6177  df1st2  6205  df2nd2  6206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-oprab 5862  df-1st 6122  df-2nd 6123
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