Theorem erinxp 7821

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 3834	. . . 4 |- ( R i^i ( B X. B ) ) C_ ( B X. B )
2		relxp 5227	. . . 4 |- Rel ( B X. B )
3		relss 5206	. . . 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 9	. . 3 |- Rel ( R i^i ( B X. B ) )
5	4	a1i 11	. 2 |- ( ph -> Rel ( R i^i ( B X. B ) ) )
6		simpr 477	. . . . 5 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> x ( R i^i ( B X. B ) ) y )
7		brinxp2 5180	. . . . 5 |- ( x ( R i^i ( B X. B ) ) y <-> ( x e. B /\ y e. B /\ x R y ) )
8	6, 7	sylib 208	. . . 4 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> ( x e. B /\ y e. B /\ x R y ) )
9	8	simp2d 1074	. . 3 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> y e. B )
10	8	simp1d 1073	. . 3 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> x e. B )
11		erinxp.r	. . . . 5 |- ( ph -> R Er A )
12	11	adantr 481	. . . 4 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> R Er A )
13	8	simp3d 1075	. . . 4 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> x R y )
14	12, 13	ersym 7754	. . 3 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> y R x )
15		brinxp2 5180	. . 3 |- ( y ( R i^i ( B X. B ) ) x <-> ( y e. B /\ x e. B /\ y R x ) )
16	9, 10, 14, 15	syl3anbrc 1246	. 2 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> y ( R i^i ( B X. B ) ) x )
17	10	adantrr 753	. . 3 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x e. B )
18		simprr 796	. . . . 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 5180	. . . . 5 |- ( y ( R i^i ( B X. B ) ) z <-> ( y e. B /\ z e. B /\ y R z ) )
20	18, 19	sylib 208	. . . 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 1074	. . 3 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> z e. B )
22	11	adantr 481	. . . 4 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> R Er A )
23	13	adantrr 753	. . . 4 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x R y )
24	20	simp3d 1075	. . . 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 7758	. . 3 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x R z )
26		brinxp2 5180	. . 3 |- ( x ( R i^i ( B X. B ) ) z <-> ( x e. B /\ z e. B /\ x R z ) )
27	17, 21, 25, 26	syl3anbrc 1246	. 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 481	. . . . . 6 |- ( ( ph /\ x e. B ) -> R Er A )
29		erinxp.a	. . . . . . 7 |- ( ph -> B C_ A )
30	29	sselda 3603	. . . . . 6 |- ( ( ph /\ x e. B ) -> x e. A )
31	28, 30	erref 7762	. . . . 5 |- ( ( ph /\ x e. B ) -> x R x )
32	31	ex 450	. . . 4 |- ( ph -> ( x e. B -> x R x ) )
33	32	pm4.71rd 667	. . 3 |- ( ph -> ( x e. B <-> ( x R x /\ x e. B ) ) )
34		brin 4704	. . . 4 |- ( x ( R i^i ( B X. B ) ) x <-> ( x R x /\ x ( B X. B ) x ) )
35		brxp 5147	. . . . . 6 |- ( x ( B X. B ) x <-> ( x e. B /\ x e. B ) )
36		anidm 676	. . . . . 6 |- ( ( x e. B /\ x e. B ) <-> x e. B )
37	35, 36	bitri 264	. . . . 5 |- ( x ( B X. B ) x <-> x e. B )
38	37	anbi2i 730	. . . 4 |- ( ( x R x /\ x ( B X. B ) x ) <-> ( x R x /\ x e. B ) )
39	34, 38	bitri 264	. . 3 |- ( x ( R i^i ( B X. B ) ) x <-> ( x R x /\ x e. B ) )
40	33, 39	syl6bbr 278	. 2 |- ( ph -> ( x e. B <-> x ( R i^i ( B X. B ) ) x ) )
41	5, 16, 27, 40	iserd 7768	1 |- ( ph -> ( R i^i ( B X. B ) ) Er B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   i^i cin 3573   C_ wss 3574   class class class wbr 4653   X. cxp 5112   Rel wrel 5119   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-er 7742

This theorem is referenced by:  frgpuplem  18185  pi1buni  22840  pi1addf  22847  pi1addval  22848  pi1grplem  22849