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Theorem dmopab 4905
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 4101 . . 3  |-  F/_ x { <. x ,  y
>.  |  ph }
2 nfopab2 4102 . . 3  |-  F/_ y { <. x ,  y
>.  |  ph }
31, 2dfdmf 4889 . 2  |-  dom  { <. x ,  y >.  |  ph }  =  {
x  |  E. y  x { <. x ,  y
>.  |  ph } y }
4 df-br 4040 . . . . 5  |-  ( x { <. x ,  y
>.  |  ph } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ph } )
5 opabid 4287 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
64, 5bitri 240 . . . 4  |-  ( x { <. x ,  y
>.  |  ph } y  <->  ph )
76exbii 1572 . . 3  |-  ( E. y  x { <. x ,  y >.  |  ph } y  <->  E. y ph )
87abbii 2408 . 2  |-  { x  |  E. y  x { <. x ,  y >.  |  ph } y }  =  { x  |  E. y ph }
93, 8eqtri 2316 1  |-  dom  { <. x ,  y >.  |  ph }  =  {
x  |  E. y ph }
Colors of variables: wff set class
Syntax hints:   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   <.cop 3656   class class class wbr 4039   {copab 4092   dom cdm 4705
This theorem is referenced by:  dmopabss  4906  dmopab3  4907  fndmin  5648  zfrep6  5764  dmoprab  5944  opabiotadm  6308  hartogslem1  7273  rankf  7482  dfac3  7764  axdc2lem  8090  shftdm  11582  adjeu  22485  mptfnf  23241  inpc  25380  dominc  25383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-dm 4715
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