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Theorem rnopab 4609
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 3855 . . 3  |-  F/_ x { <. x ,  y
>.  |  ph }
2 nfopab2 3856 . . 3  |-  F/_ y { <. x ,  y
>.  |  ph }
31, 2dfrnf 4603 . 2  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x  x { <. x ,  y
>.  |  ph } y }
4 df-br 3794 . . . . 5  |-  ( x { <. x ,  y
>.  |  ph } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ph } )
5 opabid 4020 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
64, 5bitri 182 . . . 4  |-  ( x { <. x ,  y
>.  |  ph } y  <->  ph )
76exbii 1537 . . 3  |-  ( E. x  x { <. x ,  y >.  |  ph } y  <->  E. x ph )
87abbii 2195 . 2  |-  { y  |  E. x  x { <. x ,  y
>.  |  ph } y }  =  { y  |  E. x ph }
93, 8eqtri 2102 1  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x ph }
Colors of variables: wff set class
Syntax hints:    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2068   <.cop 3409   class class class wbr 3793   {copab 3846   ran crn 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by:  rnmpt  4610  mptpreima  4844  rnoprab  5618
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