Theorem erinxp 7404

Description: A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
erinxp.r
erinxp.a

Assertion

Ref	Expression
erinxp

Proof of Theorem erinxp

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

Step	Hyp	Ref	Expression
1		inss2 3718	. . . 4
2		relxp 5115	. . . 4
3		relss 5095	. . . 4
4	1, 2, 3	mp2 9	. . 3
5	4	a1i 11	. 2
6		simpr 461	. . . . 5
7		brinxp2 5066	. . . . 5
8	6, 7	sylib 196	. . . 4
9	8	simp2d 1009	. . 3
10	8	simp1d 1008	. . 3
11		erinxp.r	. . . . 5
12	11	adantr 465	. . . 4
13	8	simp3d 1010	. . . 4
14	12, 13	ersym 7342	. . 3
15		brinxp2 5066	. . 3
16	9, 10, 14, 15	syl3anbrc 1180	. 2
17	10	adantrr 716	. . 3
18		simprr 757	. . . . 5
19		brinxp2 5066	. . . . 5
20	18, 19	sylib 196	. . . 4
21	20	simp2d 1009	. . 3
22	11	adantr 465	. . . 4
23	13	adantrr 716	. . . 4
24	20	simp3d 1010	. . . 4
25	22, 23, 24	ertrd 7346	. . 3
26		brinxp2 5066	. . 3
27	17, 21, 25, 26	syl3anbrc 1180	. 2
28	11	adantr 465	. . . . . 6
29		erinxp.a	. . . . . . 7
30	29	sselda 3503	. . . . . 6
31	28, 30	erref 7350	. . . . 5
32	31	ex 434	. . . 4
33	32	pm4.71rd 635	. . 3
34		brin 4501	. . . 4
35		brxp 5035	. . . . . 6
36		anidm 644	. . . . . 6
37	35, 36	bitri 249	. . . . 5
38	37	anbi2i 694	. . . 4
39	34, 38	bitri 249	. . 3
40	33, 39	syl6bbr 263	. 2
41	5, 16, 27, 40	iserd 7356	1

Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  e.wcel 1818  i^icin 3474  C_wss 3475   class class class wbr 4452  X.cxp 5002  Relwrel 5009  Erwer 7327

This theorem is referenced by:  frgpuplem  16790  pi1buni  21540  pi1addf  21547  pi1addval  21548  pi1grplem  21549

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-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-er 7330