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Theorem dfoprab4 6083
Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfoprab4.1  |-  ( w  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
dfoprab4  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
Distinct variable groups:    x, w, y, A    w, B, x, y    ph, x, y    ps, w    z, w, x, y
Allowed substitution hints:    ph( z, w)    ps( x, y, z)    A( z)    B( z)

Proof of Theorem dfoprab4
StepHypRef Expression
1 xpss 4642 . . . . . 6  |-  ( A  X.  B )  C_  ( _V  X.  _V )
21sseli 3088 . . . . 5  |-  ( w  e.  ( A  X.  B )  ->  w  e.  ( _V  X.  _V ) )
32adantr 274 . . . 4  |-  ( ( w  e.  ( A  X.  B )  /\  ph )  ->  w  e.  ( _V  X.  _V )
)
43pm4.71ri 389 . . 3  |-  ( ( w  e.  ( A  X.  B )  /\  ph )  <->  ( w  e.  ( _V  X.  _V )  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) )
54opabbii 3990 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) }
6 eleq1 2200 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
7 opelxp 4564 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
86, 7syl6bb 195 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
9 dfoprab4.1 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
108, 9anbi12d 464 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  ( A  X.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) ) )
1110dfoprab3 6082 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  (
w  e.  ( A  X.  B )  /\  ph ) ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
125, 11eqtri 2158 1  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2681   <.cop 3525   {copab 3983    X. cxp 4532   {coprab 5768
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-oprab 5771  df-1st 6031  df-2nd 6032
This theorem is referenced by:  dfoprab4f  6084  dfxp3  6085
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