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Theorem opabbi2dv 4794
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2308. (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 4776 . . 3  |-  ( Rel 
A  ->  { <. x ,  y >.  |  <. x ,  y >.  e.  A }  =  A )
31, 2ax-mp 5 . 2  |-  { <. x ,  y >.  |  <. x ,  y >.  e.  A }  =  A
4 opabbi2dv.3 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  ps )
)
54opabbidv 4084 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  <. x ,  y >.  e.  A }  =  { <. x ,  y >.  |  ps } )
63, 5eqtr3id 2236 1  |-  ( ph  ->  A  =  { <. x ,  y >.  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2160   <.cop 3610   {copab 4078   Rel wrel 4649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-xp 4650  df-rel 4651
This theorem is referenced by: (None)
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