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Theorem dmopabss 4746
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 4745 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  E. y ( x  e.  A  /\  ph ) }
2 19.42v 1878 . . . 4  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
32abbii 2253 . . 3  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
4 ssab2 3176 . . 3  |-  { x  |  ( x  e.  A  /\  E. y ph ) }  C_  A
53, 4eqsstri 3124 . 2  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  C_  A
61, 5eqsstri 3124 1  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103   E.wex 1468    e. wcel 1480   {cab 2123    C_ wss 3066   {copab 3983   dom cdm 4534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-dm 4544
This theorem is referenced by:  opabex  5637
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