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Theorem dfoprab4 6177
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 4793 . . . . . 6  |-  ( A  X.  B )  C_  ( _V  X.  _V )
21sseli 3176 . . . . 5  |-  ( w  e.  ( A  X.  B )  ->  w  e.  ( _V  X.  _V ) )
32adantr 451 . . . 4  |-  ( ( w  e.  ( A  X.  B )  /\  ph )  ->  w  e.  ( _V  X.  _V )
)
43pm4.71ri 614 . . 3  |-  ( ( w  e.  ( A  X.  B )  /\  ph )  <->  ( w  e.  ( _V  X.  _V )  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) )
54opabbii 4083 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) }
6 eleq1 2343 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
7 opelxp 4719 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
86, 7syl6bb 252 . . . 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 691 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  ( A  X.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) ) )
1110dfoprab3 6176 . 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 2303 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   {copab 4076    X. cxp 4687   {coprab 5859
This theorem is referenced by:  dfoprab4f  6178  dfxp3  6179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-oprab 5862  df-1st 6122  df-2nd 6123
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