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Theorem opabrn 25944
Description: Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
opabrn  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ran  R  =  { y  |  E. x ph } )
Distinct variable group:    x, y, R
Allowed substitution hints:    ph( x, y)

Proof of Theorem opabrn
StepHypRef Expression
1 dfrn2 5028 . 2  |-  ran  R  =  { y  |  E. x  x R y }
2 nfcv 2579 . . . 4  |-  F/_ y R
3 nfopab2 4359 . . . 4  |-  F/_ y { <. x ,  y
>.  |  ph }
42, 3nfeq 2586 . . 3  |-  F/ y  R  =  { <. x ,  y >.  |  ph }
5 nfcv 2579 . . . . 5  |-  F/_ x R
6 nfopab1 4358 . . . . 5  |-  F/_ x { <. x ,  y
>.  |  ph }
75, 6nfeq 2586 . . . 4  |-  F/ x  R  =  { <. x ,  y >.  |  ph }
8 df-br 4293 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
9 eleq2 2504 . . . . . 6  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( <. x ,  y >.  e.  R  <->  <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph } ) )
10 opabid 4596 . . . . . 6  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
119, 10syl6bb 261 . . . . 5  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( <. x ,  y >.  e.  R  <->  ph ) )
128, 11syl5bb 257 . . . 4  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( x R y  <->  ph ) )
137, 12exbid 1820 . . 3  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( E. x  x R y  <->  E. x ph ) )
144, 13abbid 2556 . 2  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  { y  |  E. x  x R y }  =  {
y  |  E. x ph } )
151, 14syl5eq 2487 1  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ran  R  =  { y  |  E. x ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429   <.cop 3883   class class class wbr 4292   {copab 4349   ran crn 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-cnv 4848  df-dm 4850  df-rn 4851
This theorem is referenced by:  fpwrelmapffslem  26032
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