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Theorem opabbi2dv 4598
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2207. (Contributed by NM, 24-Feb-2014.)
Hypotheses
Ref Expression
opabbi2dv.1  |-  Rel  A
opabbi2dv.3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  ps )
)
Assertion
Ref Expression
opabbi2dv  |-  ( ph  ->  A  =  { <. x ,  y >.  |  ps } )
Distinct variable groups:    x, y, A    ph, x, y
Allowed substitution hints:    ps( x, y)

Proof of Theorem opabbi2dv
StepHypRef Expression
1 opabbi2dv.1 . . 3  |-  Rel  A
2 opabid2 4580 . . 3  |-  ( Rel 
A  ->  { <. x ,  y >.  |  <. x ,  y >.  e.  A }  =  A )
31, 2ax-mp 7 . 2  |-  { <. x ,  y >.  |  <. x ,  y >.  e.  A }  =  A
4 opabbi2dv.3 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  ps )
)
54opabbidv 3910 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  <. x ,  y >.  e.  A }  =  { <. x ,  y >.  |  ps } )
63, 5syl5eqr 2135 1  |-  ( ph  ->  A  =  { <. x ,  y >.  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1290    e. wcel 1439   <.cop 3453   {copab 3904   Rel wrel 4457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-opab 3906  df-xp 4458  df-rel 4459
This theorem is referenced by: (None)
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