Theorem erinxp 6203

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	|- ( ph -> R Er A )
erinxp.a	|- ( ph -> B C_ A )

Assertion

Ref	Expression
erinxp	|- ( ph -> ( R i^i ( B X. B ) ) Er B )

Proof of Theorem erinxp

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3187	. . . 4 |- ( R i^i ( B X. B ) ) C_ ( B X. B )
2		relxp 4465	. . . 4 |- Rel ( B X. B )
3		relss 4445	. . . 4 |- ( ( R i^i ( B X. B ) ) C_ ( B X. B ) -> ( Rel ( B X. B ) -> Rel ( R i^i ( B X. B ) ) ) )
4	1, 2, 3	mp2 16	. . 3 |- Rel ( R i^i ( B X. B ) )
5	4	a1i 9	. 2 |- ( ph -> Rel ( R i^i ( B X. B ) ) )
6		simpr 108	. . . . 5 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> x ( R i^i ( B X. B ) ) y )
7		brinxp2 4425	. . . . 5 |- ( x ( R i^i ( B X. B ) ) y <-> ( x e. B /\ y e. B /\ x R y ) )
8	6, 7	sylib 120	. . . 4 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> ( x e. B /\ y e. B /\ x R y ) )
9	8	simp2d 951	. . 3 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> y e. B )
10	8	simp1d 950	. . 3 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> x e. B )
11		erinxp.r	. . . . 5 |- ( ph -> R Er A )
12	11	adantr 270	. . . 4 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> R Er A )
13	8	simp3d 952	. . . 4 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> x R y )
14	12, 13	ersym 6141	. . 3 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> y R x )
15		brinxp2 4425	. . 3 |- ( y ( R i^i ( B X. B ) ) x <-> ( y e. B /\ x e. B /\ y R x ) )
16	9, 10, 14, 15	syl3anbrc 1122	. 2 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> y ( R i^i ( B X. B ) ) x )
17	10	adantrr 462	. . 3 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x e. B )
18		simprr 498	. . . . 5 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> y ( R i^i ( B X. B ) ) z )
19		brinxp2 4425	. . . . 5 |- ( y ( R i^i ( B X. B ) ) z <-> ( y e. B /\ z e. B /\ y R z ) )
20	18, 19	sylib 120	. . . 4 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> ( y e. B /\ z e. B /\ y R z ) )
21	20	simp2d 951	. . 3 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> z e. B )
22	11	adantr 270	. . . 4 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> R Er A )
23	13	adantrr 462	. . . 4 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x R y )
24	20	simp3d 952	. . . 4 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> y R z )
25	22, 23, 24	ertrd 6145	. . 3 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x R z )
26		brinxp2 4425	. . 3 |- ( x ( R i^i ( B X. B ) ) z <-> ( x e. B /\ z e. B /\ x R z ) )
27	17, 21, 25, 26	syl3anbrc 1122	. 2 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x ( R i^i ( B X. B ) ) z )
28	11	adantr 270	. . . . . 6 |- ( ( ph /\ x e. B ) -> R Er A )
29		erinxp.a	. . . . . . 7 |- ( ph -> B C_ A )
30	29	sselda 2999	. . . . . 6 |- ( ( ph /\ x e. B ) -> x e. A )
31	28, 30	erref 6149	. . . . 5 |- ( ( ph /\ x e. B ) -> x R x )
32	31	ex 113	. . . 4 |- ( ph -> ( x e. B -> x R x ) )
33	32	pm4.71rd 386	. . 3 |- ( ph -> ( x e. B <-> ( x R x /\ x e. B ) ) )
34		brin 3832	. . . 4 |- ( x ( R i^i ( B X. B ) ) x <-> ( x R x /\ x ( B X. B ) x ) )
35		brxp 4393	. . . . . 6 |- ( x ( B X. B ) x <-> ( x e. B /\ x e. B ) )
36		anidm 388	. . . . . 6 |- ( ( x e. B /\ x e. B ) <-> x e. B )
37	35, 36	bitri 182	. . . . 5 |- ( x ( B X. B ) x <-> x e. B )
38	37	anbi2i 444	. . . 4 |- ( ( x R x /\ x ( B X. B ) x ) <-> ( x R x /\ x e. B ) )
39	34, 38	bitri 182	. . 3 |- ( x ( R i^i ( B X. B ) ) x <-> ( x R x /\ x e. B ) )
40	33, 39	syl6bbr 196	. 2 |- ( ph -> ( x e. B <-> x ( R i^i ( B X. B ) ) x ) )
41	5, 16, 27, 40	iserd 6155	1 |- ( ph -> ( R i^i ( B X. B ) ) Er B )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   e. wcel 1433   i^i cin 2972   C_ wss 2973   class class class wbr 3785   X. cxp 4361   Rel wrel 4368   Er wer 6126

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-er 6129

This theorem is referenced by: (None)