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Theorem opabdm 28058
Description: Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
opabdm  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  dom  R  =  { x  |  E. y ph } )
Distinct variable group:    x, y, R
Allowed substitution hints:    ph( x, y)

Proof of Theorem opabdm
StepHypRef Expression
1 df-dm 4864 . 2  |-  dom  R  =  { x  |  E. y  x R y }
2 nfopab1 4492 . . . 4  |-  F/_ x { <. x ,  y
>.  |  ph }
32nfeq2 2608 . . 3  |-  F/ x  R  =  { <. x ,  y >.  |  ph }
4 nfopab2 4493 . . . . 5  |-  F/_ y { <. x ,  y
>.  |  ph }
54nfeq2 2608 . . . 4  |-  F/ y  R  =  { <. x ,  y >.  |  ph }
6 df-br 4427 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
7 eleq2 2502 . . . . . 6  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( <. x ,  y >.  e.  R  <->  <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph } ) )
8 opabid 4728 . . . . . 6  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
97, 8syl6bb 264 . . . . 5  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( <. x ,  y >.  e.  R  <->  ph ) )
106, 9syl5bb 260 . . . 4  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( x R y  <->  ph ) )
115, 10exbid 1939 . . 3  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( E. y  x R y  <->  E. y ph ) )
123, 11abbid 2564 . 2  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  { x  |  E. y  x R y }  =  {
x  |  E. y ph } )
131, 12syl5eq 2482 1  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  dom  R  =  { x  |  E. y ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414   <.cop 4008   class class class wbr 4426   {copab 4483   dom cdm 4854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-dm 4864
This theorem is referenced by:  fpwrelmapffslem  28160
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