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Theorem opabss 4261
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 4259 . 2  |-  { <. x ,  y >.  |  x R y }  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  x R y ) }
2 df-br 4205 . . . . 5  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
3 eleq1 2495 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  R  <->  <. x ,  y
>.  e.  R ) )
43biimpar 472 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  <. x ,  y >.  e.  R
)  ->  z  e.  R )
52, 4sylan2b 462 . . . 4  |-  ( ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
65exlimivv 1645 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  x R y )  -> 
z  e.  R )
76abssi 3410 . 2  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  x R y ) }  C_  R
81, 7eqsstri 3370 1  |-  { <. x ,  y >.  |  x R y }  C_  R
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    C_ wss 3312   <.cop 3809   class class class wbr 4204   {copab 4257
This theorem is referenced by:  aceq3lem  7993  fullfunc  14095  fthfunc  14096  isfull  14099  isfth  14103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-in 3319  df-ss 3326  df-br 4205  df-opab 4259
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