Theorem opabss 4082

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 4080	. 2 |- { <. x , y >. | x R y } = { z | E. x E. y ( z = <. x , y >. /\ x R y ) }
2		df-br 4019	. . . . 5 |- ( x R y <-> <. x , y >. e. R )
3		eleq1 2252	. . . . . 6 |- ( z = <. x , y >. -> ( z e. R <-> <. x , y >. e. R ) )
4	3	biimpar 297	. . . . 5 |- ( ( z = <. x , y >. /\ <. x , y >. e. R ) -> z e. R )
5	2, 4	sylan2b 287	. . . 4 |- ( ( z = <. x , y >. /\ x R y ) -> z e. R )
6	5	exlimivv 1908	. . 3 |- ( E. x E. y ( z = <. x , y >. /\ x R y ) -> z e. R )
7	6	abssi 3245	. 2 |- { z | E. x E. y ( z = <. x , y >. /\ x R y ) } C_ R
8	1, 7	eqsstri 3202	1 |- { <. x , y >. | x R y } C_ R

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2160   {cab 2175   C_ wss 3144   <.cop 3610   class class class wbr 4018   {copab 4078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-in 3150  df-ss 3157  df-br 4019  df-opab 4080

This theorem is referenced by: (None)