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Theorem dfoprab3 6194
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 6193 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps ) }
2 vex 2742 . . . . . 6  |-  w  e. 
_V
3 1stexg 6170 . . . . . 6  |-  ( w  e.  _V  ->  ( 1st `  w )  e. 
_V )
42, 3ax-mp 5 . . . . 5  |-  ( 1st `  w )  e.  _V
5 2ndexg 6171 . . . . . 6  |-  ( w  e.  _V  ->  ( 2nd `  w )  e. 
_V )
62, 5ax-mp 5 . . . . 5  |-  ( 2nd `  w )  e.  _V
7 eqcom 2179 . . . . . . . . . 10  |-  ( x  =  ( 1st `  w
)  <->  ( 1st `  w
)  =  x )
8 eqcom 2179 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  w
)  <->  ( 2nd `  w
)  =  y )
97, 8anbi12i 460 . . . . . . . . 9  |-  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  <->  ( ( 1st `  w )  =  x  /\  ( 2nd `  w )  =  y ) )
10 eqopi 6175 . . . . . . . . 9  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
( 1st `  w
)  =  x  /\  ( 2nd `  w )  =  y ) )  ->  w  =  <. x ,  y >. )
119, 10sylan2b 287 . . . . . . . 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 141 . . . . . 6  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) ) )  -> 
( ps  <->  ph ) )
1514ex 115 . . . . 5  |-  ( w  e.  ( _V  X.  _V )  ->  ( ( x  =  ( 1st `  w )  /\  y  =  ( 2nd `  w
) )  ->  ( ps 
<-> 
ph ) ) )
164, 6, 15sbc2iedv 3037 . . . 4  |-  ( w  e.  ( _V  X.  _V )  ->  ( [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  / 
y ]. ps  <->  ph ) )
1716pm5.32i 454 . . 3  |-  ( ( w  e.  ( _V 
X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ps )  <->  ( w  e.  ( _V  X.  _V )  /\  ph ) )
1817opabbii 4072 . 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 2199 1  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2739   [.wsbc 2964   <.cop 3597   {copab 4065    X. cxp 4626   ` cfv 5218   {coprab 5878   1stc1st 6141   2ndc2nd 6142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226  df-oprab 5881  df-1st 6143  df-2nd 6144
This theorem is referenced by:  dfoprab4  6195  df1st2  6222  df2nd2  6223
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