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Theorem resopab 5189
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 4892 . 2  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  ( { <. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )
2 df-xp 4886 . . . . . 6  |-  ( A  X.  _V )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  _V ) }
3 vex 2961 . . . . . . . 8  |-  y  e. 
_V
43biantru 493 . . . . . . 7  |-  ( x  e.  A  <->  ( x  e.  A  /\  y  e.  _V ) )
54opabbii 4274 . . . . . 6  |-  { <. x ,  y >.  |  x  e.  A }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  _V ) }
62, 5eqtr4i 2461 . . . . 5  |-  ( A  X.  _V )  =  { <. x ,  y
>.  |  x  e.  A }
76ineq2i 3541 . . . 4  |-  ( {
<. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )  =  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  x  e.  A } )
8 incom 3535 . . . 4  |-  ( {
<. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  x  e.  A } )  =  ( { <. x ,  y
>.  |  x  e.  A }  i^i  { <. x ,  y >.  |  ph } )
97, 8eqtri 2458 . . 3  |-  ( {
<. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )  =  ( { <. x ,  y >.  |  x  e.  A }  i^i  {
<. x ,  y >.  |  ph } )
10 inopab 5007 . . 3  |-  ( {
<. x ,  y >.  |  x  e.  A }  i^i  { <. x ,  y >.  |  ph } )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
119, 10eqtri 2458 . 2  |-  ( {
<. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
121, 11eqtri 2458 1  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321   {copab 4267    X. cxp 4878    |` cres 4882
This theorem is referenced by:  resopab2  5192  opabresid  5196  mptpreima  5365  isarep2  5535  resoprab  6168  df1st2  6435  df2nd2  6436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886  df-rel 4887  df-res 4892
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