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Theorem dfoprab3 5845
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 5844 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps ) }
2 vex 2577 . . . . . 6  |-  w  e. 
_V
3 1stexg 5822 . . . . . 6  |-  ( w  e.  _V  ->  ( 1st `  w )  e. 
_V )
42, 3ax-mp 7 . . . . 5  |-  ( 1st `  w )  e.  _V
5 2ndexg 5823 . . . . . 6  |-  ( w  e.  _V  ->  ( 2nd `  w )  e. 
_V )
62, 5ax-mp 7 . . . . 5  |-  ( 2nd `  w )  e.  _V
7 eqcom 2058 . . . . . . . . . 10  |-  ( x  =  ( 1st `  w
)  <->  ( 1st `  w
)  =  x )
8 eqcom 2058 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  w
)  <->  ( 2nd `  w
)  =  y )
97, 8anbi12i 441 . . . . . . . . 9  |-  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  <->  ( ( 1st `  w )  =  x  /\  ( 2nd `  w )  =  y ) )
10 eqopi 5826 . . . . . . . . 9  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
( 1st `  w
)  =  x  /\  ( 2nd `  w )  =  y ) )  ->  w  =  <. x ,  y >. )
119, 10sylan2b 275 . . . . . . . 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 133 . . . . . 6  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  -> 
( ps  <->  ph ) )
1514ex 112 . . . . 5  |-  ( w  e.  ( _V  X.  _V )  ->  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  ->  ( ps 
<-> 
ph ) ) )
164, 6, 15sbc2iedv 2858 . . . 4  |-  ( w  e.  ( _V  X.  _V )  ->  ( [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  / 
y ]. ps  <->  ph ) )
1716pm5.32i 435 . . 3  |-  ( ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps )  <->  ( w  e.  ( _V  X.  _V )  /\  ph ) )
1817opabbii 3852 . 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 2077 1  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   [.wsbc 2787   <.cop 3406   {copab 3845    X. cxp 4371   ` cfv 4930   {coprab 5541   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-oprab 5544  df-1st 5795  df-2nd 5796
This theorem is referenced by:  dfoprab4  5846  df1st2  5868  df2nd2  5869
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