Theorem opabss 2673

Description: The collection of ordered pairs in a class is a subclass of it.

Assertion

Ref	Expression
opabss	|- {<.x, y>. | xRy} (_ R

Distinct variable groups:   x,R   y,R

Proof of Theorem opabss

Step	Hyp	Ref	Expression
1		df-opab 2672	. . 3 |- {<.x, y>. | xRy} = {z | E.xE.y(z = <.x, y>. /\ xRy)}
2		eleq1 1537	. . . . . . 7 |- (z = <.x, y>. -> (z e. R <-> <.x, y>. e. R))
3	2	biimpar 419	. . . . . 6 |- ((z = <.x, y>. /\ <.x, y>. e. R) -> z e. R)
4		df-br 2625	. . . . . 6 |- (xRy <-> <.x, y>. e. R)
5	3, 4	sylan2b 454	. . . . 5 |- ((z = <.x, y>. /\ xRy) -> z e. R)
6	5	19.23aivv 1298	. . . 4 |- (E.xE.y(z = <.x, y>. /\ xRy) -> z e. R)
7	6	ss2abi 2123	. . 3 |- {z | E.xE.y(z = <.x, y>. /\ xRy)} (_ {z | z e. R}
8	1, 7	eqsstr 2094	. 2 |- {<.x, y>. | xRy} (_ {z | z e. R}
9		abid2 1583	. 2 |- {z | z e. R} = R
10	8, 9	sseqtr 2096	1 |- {<.x, y>. | xRy} (_ R

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  {cab 1466   (_ wss 2050  <.cop 2415   class class class wbr 2624  {copab 2671

This theorem is referenced by:  cotr 3442  cnvsym 3443

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056  df-br 2625  df-opab 2672

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