Theorem opabss 4046

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.

Step	Hyp	Ref	Expression
1		df-opab 4044	. 2 |- { <. x , y >. | x R y } = { z | E. x E. y ( z = <. x , y >. /\ x R y ) }
2		df-br 3983	. . . . 5 |- ( x R y <-> <. x , y >. e. R )
3		eleq1 2229	. . . . . 6 |- ( z = <. x , y >. -> ( z e. R <-> <. x , y >. e. R ) )
4	3	biimpar 295	. . . . 5 |- ( ( z = <. x , y >. /\ <. x , y >. e. R ) -> z e. R )
5	2, 4	sylan2b 285	. . . 4 |- ( ( z = <. x , y >. /\ x R y ) -> z e. R )
6	5	exlimivv 1884	. . 3 |- ( E. x E. y ( z = <. x , y >. /\ x R y ) -> z e. R )
7	6	abssi 3217	. 2 |- { z | E. x E. y ( z = <. x , y >. /\ x R y ) } C_ R
8	1, 7	eqsstri 3174	1 |- { <. x , y >. | x R y } C_ R

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   C_ wss 3116   <.cop 3579   class class class wbr 3982   {copab 4042

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-br 3983  df-opab 4044

This theorem is referenced by: (None)