MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmopab Unicode version

Theorem dmopab 4889
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 4085 . . 3  |-  F/_ x { <. x ,  y
>.  |  ph }
2 nfopab2 4086 . . 3  |-  F/_ y { <. x ,  y
>.  |  ph }
31, 2dfdmf 4873 . 2  |-  dom  { <. x ,  y >.  |  ph }  =  {
x  |  E. y  x { <. x ,  y
>.  |  ph } y }
4 df-br 4024 . . . . 5  |-  ( x { <. x ,  y
>.  |  ph } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ph } )
5 opabid 4271 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
64, 5bitri 240 . . . 4  |-  ( x { <. x ,  y
>.  |  ph } y  <->  ph )
76exbii 1569 . . 3  |-  ( E. y  x { <. x ,  y >.  |  ph } y  <->  E. y ph )
87abbii 2395 . 2  |-  { x  |  E. y  x { <. x ,  y >.  |  ph } y }  =  { x  |  E. y ph }
93, 8eqtri 2303 1  |-  dom  { <. x ,  y >.  |  ph }  =  {
x  |  E. y ph }
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   <.cop 3643   class class class wbr 4023   {copab 4076   dom cdm 4689
This theorem is referenced by:  dmopabss  4890  dmopab3  4891  fndmin  5632  zfrep6  5748  dmoprab  5928  opabiotadm  6292  hartogslem1  7257  rankf  7466  dfac3  7748  axdc2lem  8074  shftdm  11566  adjeu  22469  inpc  24689  dominc  24692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-dm 4699
  Copyright terms: Public domain W3C validator