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Theorem dfoprab4 6193
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 4809 . . . . . 6  |-  ( A  X.  B )  C_  ( _V  X.  _V )
21sseli 3189 . . . . 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 4099 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) }
6 eleq1 2356 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
7 opelxp 4735 . . . . 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 6192 . 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 2316 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   {copab 4092    X. cxp 4703   {coprab 5875
This theorem is referenced by:  dfoprab4f  6194  dfxp3  6195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-oprab 5878  df-1st 6138  df-2nd 6139
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