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Theorem opabrn 27135
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 5189 . 2  |-  ran  R  =  { y  |  E. x  x R y }
2 nfcv 2629 . . . 4  |-  F/_ y R
3 nfopab2 4514 . . . 4  |-  F/_ y { <. x ,  y
>.  |  ph }
42, 3nfeq 2640 . . 3  |-  F/ y  R  =  { <. x ,  y >.  |  ph }
5 nfcv 2629 . . . . 5  |-  F/_ x R
6 nfopab1 4513 . . . . 5  |-  F/_ x { <. x ,  y
>.  |  ph }
75, 6nfeq 2640 . . . 4  |-  F/ x  R  =  { <. x ,  y >.  |  ph }
8 df-br 4448 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
9 eleq2 2540 . . . . . 6  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( <. x ,  y >.  e.  R  <->  <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph } ) )
10 opabid 4754 . . . . . 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 1834 . . 3  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( E. x  x R y  <->  E. x ph ) )
144, 13abbid 2602 . 2  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  { y  |  E. x  x R y }  =  {
y  |  E. x ph } )
151, 14syl5eq 2520 1  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ran  R  =  { y  |  E. x ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   <.cop 4033   class class class wbr 4447   {copab 4504   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  fpwrelmapffslem  27224
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