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Theorem opabbi2dv 5025
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2553. (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 5007 . . 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 4274 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  <. x ,  y >.  e.  A }  =  { <. x ,  y >.  |  ps } )
63, 5syl5eqr 2484 1  |-  ( ph  ->  A  =  { <. x ,  y >.  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   <.cop 3819   {copab 4268   Rel wrel 4886
This theorem is referenced by:  recmulnq  8846  dib1dim  32037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270  df-xp 4887  df-rel 4888
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