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Theorem resopab 4996
Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
Assertion
Ref Expression
resopab  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem resopab
StepHypRef Expression
1 df-res 4701 . 2  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  ( { <. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )
2 df-xp 4695 . . . . . 6  |-  ( A  X.  _V )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  _V ) }
3 vex 2791 . . . . . . . 8  |-  y  e. 
_V
43biantru 491 . . . . . . 7  |-  ( x  e.  A  <->  ( x  e.  A  /\  y  e.  _V ) )
54opabbii 4083 . . . . . 6  |-  { <. x ,  y >.  |  x  e.  A }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  _V ) }
62, 5eqtr4i 2306 . . . . 5  |-  ( A  X.  _V )  =  { <. x ,  y
>.  |  x  e.  A }
76ineq2i 3367 . . . 4  |-  ( {
<. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )  =  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  x  e.  A } )
8 incom 3361 . . . 4  |-  ( {
<. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  x  e.  A } )  =  ( { <. x ,  y
>.  |  x  e.  A }  i^i  { <. x ,  y >.  |  ph } )
97, 8eqtri 2303 . . 3  |-  ( {
<. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )  =  ( { <. x ,  y >.  |  x  e.  A }  i^i  {
<. x ,  y >.  |  ph } )
10 inopab 4816 . . 3  |-  ( {
<. x ,  y >.  |  x  e.  A }  i^i  { <. x ,  y >.  |  ph } )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
119, 10eqtri 2303 . 2  |-  ( {
<. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
121, 11eqtri 2303 1  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   {copab 4076    X. cxp 4687    |` cres 4691
This theorem is referenced by:  resopab2  4999  opabresid  5003  mptpreima  5166  isarep2  5332  resoprab  5940  df1st2  6205  df2nd2  6206
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-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-opab 4078  df-xp 4695  df-rel 4696  df-res 4701
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