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Theorem dfoprab3 6395
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 6394 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps ) }
2 fvex 5734 . . . . 5  |-  ( 1st `  w )  e.  _V
3 fvex 5734 . . . . 5  |-  ( 2nd `  w )  e.  _V
4 eqcom 2437 . . . . . . . . . 10  |-  ( x  =  ( 1st `  w
)  <->  ( 1st `  w
)  =  x )
5 eqcom 2437 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  w
)  <->  ( 2nd `  w
)  =  y )
64, 5anbi12i 679 . . . . . . . . 9  |-  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  <->  ( ( 1st `  w )  =  x  /\  ( 2nd `  w )  =  y ) )
7 eqopi 6375 . . . . . . . . 9  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
( 1st `  w
)  =  x  /\  ( 2nd `  w )  =  y ) )  ->  w  =  <. x ,  y >. )
86, 7sylan2b 462 . . . . . . . 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 16 . . . . . . 7  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  -> 
( ph  <->  ps ) )
1110bicomd 193 . . . . . 6  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  -> 
( ps  <->  ph ) )
1211ex 424 . . . . 5  |-  ( w  e.  ( _V  X.  _V )  ->  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  ->  ( ps 
<-> 
ph ) ) )
132, 3, 12sbc2iedv 3221 . . . 4  |-  ( w  e.  ( _V  X.  _V )  ->  ( [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  / 
y ]. ps  <->  ph ) )
1413pm5.32i 619 . . 3  |-  ( ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps )  <->  ( w  e.  ( _V  X.  _V )  /\  ph ) )
1514opabbii 4264 . 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 2456 1  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   [.wsbc 3153   <.cop 3809   {copab 4257    X. cxp 4868   ` cfv 5446   {coprab 6074   1stc1st 6339   2ndc2nd 6340
This theorem is referenced by:  dfoprab4  6396  df1st2  6425  df2nd2  6426
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-oprab 6077  df-1st 6341  df-2nd 6342
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