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Theorem rnopab 4940
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 4101 . . 3  |-  F/_ x { <. x ,  y
>.  |  ph }
2 nfopab2 4102 . . 3  |-  F/_ y { <. x ,  y
>.  |  ph }
31, 2dfrnf 4933 . 2  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x  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. x  x { <. x ,  y >.  |  ph } y  <->  E. x ph )
87abbii 2408 . 2  |-  { y  |  E. x  x { <. x ,  y
>.  |  ph } y }  =  { y  |  E. x ph }
93, 8eqtri 2316 1  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x 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   ran crn 4706
This theorem is referenced by:  rnmpt  4941  mptpreima  5182  rnoprab  5946  marypha2lem4  7207  hartogslem1  7273  axdc2lem  8090  abrexdomjm  23181  abrexexd  23207  rninc  25384  abrexdom  26508
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-cnv 4713  df-dm 4715  df-rn 4716
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