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Theorem opabss 3924
Description: The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
opabss  |-  { <. x ,  y >.  |  x R y }  C_  R
Distinct variable groups:    x, R    y, R

Proof of Theorem opabss
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-opab 3922 . 2  |-  { <. x ,  y >.  |  x R y }  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  x R y ) }
2 df-br 3868 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
3 eleq1 2157 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  R  <->  <. x ,  y
>.  e.  R ) )
43biimpar 292 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  <. x ,  y >.  e.  R
)  ->  z  e.  R )
52, 4sylan2b 282 . . . 4  |-  ( ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
65exlimivv 1831 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
76abssi 3111 . 2  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  x R y ) }  C_  R
81, 7eqsstri 3071 1  |-  { <. x ,  y >.  |  x R y }  C_  R
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1296   E.wex 1433    e. wcel 1445   {cab 2081    C_ wss 3013   <.cop 3469   class class class wbr 3867   {copab 3920
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-in 3019  df-ss 3026  df-br 3868  df-opab 3922
This theorem is referenced by: (None)
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