Theorem opabss 4714

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 4713	. 2 |- { <. x , y >. | x R y } = { z | E. x E. y ( z = <. x , y >. /\ x R y ) }
2		df-br 4654	. . . . 5 |- ( x R y <-> <. x , y >. e. R )
3		eleq1 2689	. . . . . 6 |- ( z = <. x , y >. -> ( z e. R <-> <. x , y >. e. R ) )
4	3	biimpar 502	. . . . 5 |- ( ( z = <. x , y >. /\ <. x , y >. e. R ) -> z e. R )
5	2, 4	sylan2b 492	. . . 4 |- ( ( z = <. x , y >. /\ x R y ) -> z e. R )
6	5	exlimivv 1860	. . 3 |- ( E. x E. y ( z = <. x , y >. /\ x R y ) -> z e. R )
7	6	abssi 3677	. 2 |- { z | E. x E. y ( z = <. x , y >. /\ x R y ) } C_ R
8	1, 7	eqsstri 3635	1 |- { <. x , y >. | x R y } C_ R

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   C_ wss 3574   <.cop 4183   class class class wbr 4653   {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713

This theorem is referenced by:  aceq3lem  8943  fullfunc  16566  fthfunc  16567  isfull  16570  isfth  16574