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Theorem dfoprab4 6404
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 4982 . . . . . 6  |-  ( A  X.  B )  C_  ( _V  X.  _V )
21sseli 3344 . . . . 5  |-  ( w  e.  ( A  X.  B )  ->  w  e.  ( _V  X.  _V ) )
32adantr 452 . . . 4  |-  ( ( w  e.  ( A  X.  B )  /\  ph )  ->  w  e.  ( _V  X.  _V )
)
43pm4.71ri 615 . . 3  |-  ( ( w  e.  ( A  X.  B )  /\  ph )  <->  ( w  e.  ( _V  X.  _V )  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) )
54opabbii 4272 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) }
6 eleq1 2496 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
7 opelxp 4908 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
86, 7syl6bb 253 . . . 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 692 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  ( A  X.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) ) )
1110dfoprab3 6403 . 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 2456 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   {copab 4265    X. cxp 4876   {coprab 6082
This theorem is referenced by:  dfoprab4f  6405  dfxp3  6406
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-oprab 6085  df-1st 6349  df-2nd 6350
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