MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnopab Structured version   Unicode version

Theorem rnopab 5107
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 4266 . . 3  |-  F/_ x { <. x ,  y
>.  |  ph }
2 nfopab2 4267 . . 3  |-  F/_ y { <. x ,  y
>.  |  ph }
31, 2dfrnf 5100 . 2  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x  x { <. x ,  y
>.  |  ph } y }
4 df-br 4205 . . . . 5  |-  ( x { <. x ,  y
>.  |  ph } y  <->  <. x ,  y >.  e.  { <. x ,  y
>.  |  ph } )
5 opabid 4453 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
64, 5bitri 241 . . . 4  |-  ( x { <. x ,  y
>.  |  ph } y  <->  ph )
76exbii 1592 . . 3  |-  ( E. x  x { <. x ,  y >.  |  ph } y  <->  E. x ph )
87abbii 2547 . 2  |-  { y  |  E. x  x { <. x ,  y
>.  |  ph } y }  =  { y  |  E. x ph }
93, 8eqtri 2455 1  |-  ran  { <. x ,  y >.  |  ph }  =  {
y  |  E. x ph }
Colors of variables: wff set class
Syntax hints:   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   <.cop 3809   class class class wbr 4204   {copab 4257   ran crn 4871
This theorem is referenced by:  rnmpt  5108  mptpreima  5355  rnoprab  6148  marypha2lem4  7435  hartogslem1  7503  axdc2lem  8320  abrexdomjm  23980  abrexexd  23982  abrexdom  26423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-cnv 4878  df-dm 4880  df-rn 4881
  Copyright terms: Public domain W3C validator