Theorem opabss 2668

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 2667	. . 3 |- {<.x, y>. | xRy} = {z | E.xE.y(z = <.x, y>. /\ xRy)}
2		eleq1 1534	. . . . . . 7 |- (z = <.x, y>. -> (z e. R <-> <.x, y>. e. R))
3	2	biimpar 417	. . . . . 6 |- ((z = <.x, y>. /\ <.x, y>. e. R) -> z e. R)
4		df-br 2620	. . . . . 6 |- (xRy <-> <.x, y>. e. R)
5	3, 4	sylan2b 452	. . . . 5 |- ((z = <.x, y>. /\ xRy) -> z e. R)
6	5	19.23aivv 1296	. . . 4 |- (E.xE.y(z = <.x, y>. /\ xRy) -> z e. R)
7	6	ss2abi 2120	. . 3 |- {z | E.xE.y(z = <.x, y>. /\ xRy)} (_ {z | z e. R}
8	1, 7	eqsstr 2091	. 2 |- {<.x, y>. | xRy} (_ {z | z e. R}
9		abid2 1580	. 2 |- {z | z e. R} = R
10	8, 9	sseqtr 2093	1 |- {<.x, y>. | xRy} (_ R

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   (_ wss 2047  <.cop 2411   class class class wbr 2619  {copab 2666

This theorem is referenced by:  cotr 3436  cnvsym 3437

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053  df-br 2620  df-opab 2667

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