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Theorem dfoprab4 6756
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 5022 . . . . . 6  |-  ( A  X.  B )  C_  ( _V  X.  _V )
21sseli 3413 . . . . 5  |-  ( w  e.  ( A  X.  B )  ->  w  e.  ( _V  X.  _V ) )
32adantr 463 . . . 4  |-  ( ( w  e.  ( A  X.  B )  /\  ph )  ->  w  e.  ( _V  X.  _V )
)
43pm4.71ri 631 . . 3  |-  ( ( w  e.  ( A  X.  B )  /\  ph )  <->  ( w  e.  ( _V  X.  _V )  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) )
54opabbii 4431 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  ( w  e.  ( A  X.  B
)  /\  ph ) ) }
6 eleq1 2454 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
7 opelxp 4943 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
86, 7syl6bb 261 . . . 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 708 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  ( A  X.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) ) )
1110dfoprab3 6755 . 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 2411 1  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   <.cop 3950   {copab 4424    X. cxp 4911   {coprab 6197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fv 5504  df-oprab 6200  df-1st 6699  df-2nd 6700
This theorem is referenced by:  dfoprab4f  6757  dfxp3  6759
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