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Theorem resopab 5330
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 5020 . 2  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  ( { <. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )
2 df-xp 5014 . . . . . 6  |-  ( A  X.  _V )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  _V ) }
3 vex 3112 . . . . . . . 8  |-  y  e. 
_V
43biantru 505 . . . . . . 7  |-  ( x  e.  A  <->  ( x  e.  A  /\  y  e.  _V ) )
54opabbii 4521 . . . . . 6  |-  { <. x ,  y >.  |  x  e.  A }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  _V ) }
62, 5eqtr4i 2489 . . . . 5  |-  ( A  X.  _V )  =  { <. x ,  y
>.  |  x  e.  A }
76ineq2i 3693 . . . 4  |-  ( {
<. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )  =  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  x  e.  A } )
8 incom 3687 . . . 4  |-  ( {
<. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  x  e.  A } )  =  ( { <. x ,  y
>.  |  x  e.  A }  i^i  { <. x ,  y >.  |  ph } )
97, 8eqtri 2486 . . 3  |-  ( {
<. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )  =  ( { <. x ,  y >.  |  x  e.  A }  i^i  {
<. x ,  y >.  |  ph } )
10 inopab 5143 . . 3  |-  ( {
<. x ,  y >.  |  x  e.  A }  i^i  { <. x ,  y >.  |  ph } )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
119, 10eqtri 2486 . 2  |-  ( {
<. x ,  y >.  |  ph }  i^i  ( A  X.  _V ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
121, 11eqtri 2486 1  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470   {copab 4514    X. cxp 5006    |` cres 5010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516  df-xp 5014  df-rel 5015  df-res 5020
This theorem is referenced by:  resopab2  5332  opabresid  5337  mptpreima  5506  isarep2  5674  resoprab  6397  elrnmpt2res  6415  df1st2  6885  df2nd2  6886
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