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Theorem opabss 3844
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 3842 . 2  |-  { <. x ,  y >.  |  x R y }  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  x R y ) }
2 df-br 3788 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
3 eleq1 2142 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  R  <->  <. x ,  y
>.  e.  R ) )
43biimpar 291 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  <. x ,  y >.  e.  R
)  ->  z  e.  R )
52, 4sylan2b 281 . . . 4  |-  ( ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
65exlimivv 1818 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
76abssi 3070 . 2  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  x R y ) }  C_  R
81, 7eqsstri 3030 1  |-  { <. x ,  y >.  |  x R y }  C_  R
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2068    C_ wss 2974   <.cop 3403   class class class wbr 3787   {copab 3840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987  df-br 3788  df-opab 3842
This theorem is referenced by: (None)
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