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Theorem dmopabss 5151
Description: Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
Assertion
Ref Expression
dmopabss  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem dmopabss
StepHypRef Expression
1 dmopab 5150 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  E. y ( x  e.  A  /\  ph ) }
2 19.42v 1933 . . . 4  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
32abbii 2585 . . 3  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
4 ssab2 3536 . . 3  |-  { x  |  ( x  e.  A  /\  E. y ph ) }  C_  A
53, 4eqsstri 3486 . 2  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  C_  A
61, 5eqsstri 3486 1  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   E.wex 1587    e. wcel 1758   {cab 2436    C_ wss 3428   {copab 4449   dom cdm 4940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-dm 4950
This theorem is referenced by:  fvopab4ndm  5895  opabex  6047  perpln1  23231  dmadjss  25428  abrexdomjm  26025  abrexdom  28764  clwwlknprop  30575
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