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Theorem opabss 3821
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 3819 . 2  |-  { <. x ,  y >.  |  x R y }  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  x R y ) }
2 df-br 3765 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
3 eleq1 2100 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  R  <->  <. x ,  y
>.  e.  R ) )
43biimpar 281 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  <. x ,  y >.  e.  R
)  ->  z  e.  R )
52, 4sylan2b 271 . . . 4  |-  ( ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
65exlimivv 1776 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
76abssi 3015 . 2  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  x R y ) }  C_  R
81, 7eqsstri 2975 1  |-  { <. x ,  y >.  |  x R y }  C_  R
Colors of variables: wff set class
Syntax hints:    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   {cab 2026    C_ wss 2917   <.cop 3378   class class class wbr 3764   {copab 3817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-in 2924  df-ss 2931  df-br 3765  df-opab 3819
This theorem is referenced by: (None)
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