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Theorem dmopabss 5035
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 5034 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  E. y ( x  e.  A  /\  ph ) }
2 19.42v 1799 . . . 4  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
32abbii 2536 . . 3  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
4 ssab2 3523 . . 3  |-  { x  |  ( x  e.  A  /\  E. y ph ) }  C_  A
53, 4eqsstri 3472 . 2  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  C_  A
61, 5eqsstri 3472 1  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367   E.wex 1633    e. wcel 1842   {cab 2387    C_ wss 3414   {copab 4452   dom cdm 4823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-dm 4833
This theorem is referenced by:  fvopab4ndm  5956  opabex  6122  perpln1  24473  clwwlknprop  25189  dmadjss  27219  abrexdomjm  27820  abrexdom  31503
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