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Theorem rnopab 5057
Description: The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
rnopab  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x ph }
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem rnopab
StepHypRef Expression
1 nfopab1 4217 . . 3  |-  F/_ x { <. x ,  y
>.  |  ph }
2 nfopab2 4218 . . 3  |-  F/_ y { <. x ,  y
>.  |  ph }
31, 2dfrnf 5050 . 2  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x  x { <. x ,  y
>.  |  ph } y }
4 df-br 4156 . . . . 5  |-  ( x { <. x ,  y
>.  |  ph } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ph } )
5 opabid 4404 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
64, 5bitri 241 . . . 4  |-  ( x { <. x ,  y
>.  |  ph } y  <->  ph )
76exbii 1589 . . 3  |-  ( E. x  x { <. x ,  y >.  |  ph } y  <->  E. x ph )
87abbii 2501 . 2  |-  { y  |  E. x  x { <. x ,  y
>.  |  ph } y }  =  { y  |  E. x ph }
93, 8eqtri 2409 1  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x ph }
Colors of variables: wff set class
Syntax hints:   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2375   <.cop 3762   class class class wbr 4155   {copab 4208   ran crn 4821
This theorem is referenced by:  rnmpt  5058  mptpreima  5305  rnoprab  6097  marypha2lem4  7380  hartogslem1  7446  axdc2lem  8263  abrexdomjm  23834  abrexexd  23836  abrexdom  26125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-cnv 4828  df-dm 4830  df-rn 4831
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