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Theorem dmopabss 5052
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 5051 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  E. y ( x  e.  A  /\  ph ) }
2 19.42v 1842 . . . 4  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
32abbii 2587 . . 3  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
4 ssab2 3499 . . 3  |-  { x  |  ( x  e.  A  /\  E. y ph ) }  C_  A
53, 4eqsstri 3448 . 2  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  C_  A
61, 5eqsstri 3448 1  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376   E.wex 1671    e. wcel 1904   {cab 2457    C_ wss 3390   {copab 4453   dom cdm 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-dm 4849
This theorem is referenced by:  fvopab4ndm  5987  opabex  6150  perpln1  24834  clwwlknprop  25579  dmadjss  27621  abrexdomjm  28220  abrexdom  32121
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