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Theorem rnopab 4876
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 4074 . . 3  |-  F/_ x { <. x ,  y
>.  |  ph }
2 nfopab2 4075 . . 3  |-  F/_ y { <. x ,  y
>.  |  ph }
31, 2dfrnf 4870 . 2  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x  x { <. x ,  y
>.  |  ph } y }
4 df-br 4006 . . . . 5  |-  ( x { <. x ,  y
>.  |  ph } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ph } )
5 opabid 4259 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
64, 5bitri 184 . . . 4  |-  ( x { <. x ,  y
>.  |  ph } y  <->  ph )
76exbii 1605 . . 3  |-  ( E. x  x { <. x ,  y >.  |  ph } y  <->  E. x ph )
87abbii 2293 . 2  |-  { y  |  E. x  x { <. x ,  y
>.  |  ph } y }  =  { y  |  E. x ph }
93, 8eqtri 2198 1  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x ph }
Colors of variables: wff set class
Syntax hints:    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   <.cop 3597   class class class wbr 4005   {copab 4065   ran crn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-cnv 4636  df-dm 4638  df-rn 4639
This theorem is referenced by:  rnmpt  4877  mptpreima  5124  rnoprab  5960
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