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Theorem dmopab 5080
Description: The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
dmopab  |-  dom  { <. x ,  y >.  |  ph }  =  {
x  |  E. y ph }
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem dmopab
StepHypRef Expression
1 nfopab1 4274 . . 3  |-  F/_ x { <. x ,  y
>.  |  ph }
2 nfopab2 4275 . . 3  |-  F/_ y { <. x ,  y
>.  |  ph }
31, 2dfdmf 5064 . 2  |-  dom  { <. x ,  y >.  |  ph }  =  {
x  |  E. y  x { <. x ,  y
>.  |  ph } y }
4 df-br 4213 . . . . 5  |-  ( x { <. x ,  y
>.  |  ph } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ph } )
5 opabid 4461 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
64, 5bitri 241 . . . 4  |-  ( x { <. x ,  y
>.  |  ph } y  <->  ph )
76exbii 1592 . . 3  |-  ( E. y  x { <. x ,  y >.  |  ph } y  <->  E. y ph )
87abbii 2548 . 2  |-  { x  |  E. y  x { <. x ,  y >.  |  ph } y }  =  { x  |  E. y ph }
93, 8eqtri 2456 1  |-  dom  { <. x ,  y >.  |  ph }  =  {
x  |  E. y ph }
Colors of variables: wff set class
Syntax hints:   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422   <.cop 3817   class class class wbr 4212   {copab 4265   dom cdm 4878
This theorem is referenced by:  dmopabss  5081  dmopab3  5082  fndmin  5837  zfrep6  5968  dmoprab  6154  opabiotadm  6537  hartogslem1  7511  rankf  7720  dfac3  8002  axdc2lem  8328  shftdm  11886  adjeu  23392  mptfnf  24073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-dm 4888
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