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Theorem opabss 4053
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 4051 . 2  |-  { <. x ,  y >.  |  x R y }  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  x R y ) }
2 df-br 3990 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
3 eleq1 2233 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  R  <->  <. x ,  y
>.  e.  R ) )
43biimpar 295 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  <. x ,  y >.  e.  R
)  ->  z  e.  R )
52, 4sylan2b 285 . . . 4  |-  ( ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
65exlimivv 1889 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
76abssi 3222 . 2  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  x R y ) }  C_  R
81, 7eqsstri 3179 1  |-  { <. x ,  y >.  |  x R y }  C_  R
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348   E.wex 1485    e. wcel 2141   {cab 2156    C_ wss 3121   <.cop 3586   class class class wbr 3989   {copab 4049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-br 3990  df-opab 4051
This theorem is referenced by: (None)
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