MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opabbi2dv Unicode version

Theorem opabbi2dv 4981
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2519. (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 4963 . . 3  |-  ( Rel 
A  ->  { <. x ,  y >.  |  <. x ,  y >.  e.  A }  =  A )
31, 2ax-mp 8 . 2  |-  { <. x ,  y >.  |  <. x ,  y >.  e.  A }  =  A
4 opabbi2dv.3 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  ps )
)
54opabbidv 4231 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  <. x ,  y >.  e.  A }  =  { <. x ,  y >.  |  ps } )
63, 5syl5eqr 2450 1  |-  ( ph  ->  A  =  { <. x ,  y >.  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   <.cop 3777   {copab 4225   Rel wrel 4842
This theorem is referenced by:  recmulnq  8797  dib1dim  31648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-xp 4843  df-rel 4844
  Copyright terms: Public domain W3C validator