Theorem opabid2 5137

Description: A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)

Assertion

Ref	Expression
opabid2

Distinct variable group:   ,,

Proof of Theorem opabid2

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3112	. . . 4
2		vex 3112	. . . 4
3		opeq1 4217	. . . . 5
4	3	eleq1d 2526	. . . 4
5		opeq2 4218	. . . . 5
6	5	eleq1d 2526	. . . 4
7	1, 2, 4, 6	opelopab 4774	. . 3
8	7	gen2 1619	. 2
9		relopab 5134	. . 3
10		eqrel 5097	. . 3
11	9, 10	mpan 670	. 2
12	8, 11	mpbiri 233	1

Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  e.wcel 1818  <.cop 4035  {copab 4509  Relwrel 5009

This theorem is referenced by:  opabbi2dv  5157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-xp 5010  df-rel 5011