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Theorem opabrn 27902
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 5012 . 2  |-  ran  R  =  { y  |  E. x  x R y }
2 nfopab2 4462 . . . 4  |-  F/_ y { <. x ,  y
>.  |  ph }
32nfeq2 2581 . . 3  |-  F/ y  R  =  { <. x ,  y >.  |  ph }
4 nfopab1 4461 . . . . 5  |-  F/_ x { <. x ,  y
>.  |  ph }
54nfeq2 2581 . . . 4  |-  F/ x  R  =  { <. x ,  y >.  |  ph }
6 df-br 4396 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
7 eleq2 2475 . . . . . 6  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( <. x ,  y >.  e.  R  <->  <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph } ) )
8 opabid 4697 . . . . . 6  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
97, 8syl6bb 261 . . . . 5  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( <. x ,  y >.  e.  R  <->  ph ) )
106, 9syl5bb 257 . . . 4  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( x R y  <->  ph ) )
115, 10exbid 1910 . . 3  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( E. x  x R y  <->  E. x ph ) )
123, 11abbid 2537 . 2  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  { y  |  E. x  x R y }  =  {
y  |  E. x ph } )
131, 12syl5eq 2455 1  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ran  R  =  { y  |  E. x ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387   <.cop 3978   class class class wbr 4395   {copab 4452   ran crn 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-cnv 4831  df-dm 4833  df-rn 4834
This theorem is referenced by:  fpwrelmapffslem  28002
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