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Theorem dmopabss 4575
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 4574 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  E. y ( x  e.  A  /\  ph ) }
2 19.42v 1828 . . . 4  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
32abbii 2195 . . 3  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
4 ssab2 3079 . . 3  |-  { x  |  ( x  e.  A  /\  E. y ph ) }  C_  A
53, 4eqsstri 3030 . 2  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  C_  A
61, 5eqsstri 3030 1  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
Colors of variables: wff set class
Syntax hints:    /\ wa 102   E.wex 1422    e. wcel 1434   {cab 2068    C_ wss 2974   {copab 3846   dom cdm 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-dm 4381
This theorem is referenced by:  opabex  5417
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