Theorem opabss 4179

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 4177	. 2 |- { <. x , y >. | x R y } = { z | E. x E. y ( z = <. x , y >. /\ x R y ) }
2		df-br 4115	. . . . 5 |- ( x R y <-> <. x , y >. e. R )
3		eleq1 2297	. . . . . 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 1948	. . 3 |- ( E. x E. y ( z = <. x , y >. /\ x R y ) -> z e. R )
7	6	abssi 3317	. 2 |- { z | E. x E. y ( z = <. x , y >. /\ x R y ) } C_ R
8	1, 7	eqsstri 3274	1 |- { <. x , y >. | x R y } C_ R

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   {cab 2220   C_ wss 3214   <.cop 3697   class class class wbr 4114   {copab 4175

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-in 3220  df-ss 3227  df-br 4115  df-opab 4177

This theorem is referenced by:  elopabr  4406  elopabran  4407  trlsex  16508  eupthsg  16566