Theorem opabss 4097

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 4095	. 2 |- { <. x , y >. | x R y } = { z | E. x E. y ( z = <. x , y >. /\ x R y ) }
2		df-br 4034	. . . . 5 |- ( x R y <-> <. x , y >. e. R )
3		eleq1 2259	. . . . . 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 1911	. . 3 |- ( E. x E. y ( z = <. x , y >. /\ x R y ) -> z e. R )
7	6	abssi 3258	. 2 |- { z | E. x E. y ( z = <. x , y >. /\ x R y ) } C_ R
8	1, 7	eqsstri 3215	1 |- { <. x , y >. | x R y } C_ R

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   {cab 2182   C_ wss 3157   <.cop 3625   class class class wbr 4033   {copab 4093

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-br 4034  df-opab 4095

This theorem is referenced by: (None)