Theorem erinxp 6821

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 3430	. . . 4 |- ( R i^i ( B X. B ) ) C_ ( B X. B )
2		relxp 4841	. . . 4 |- Rel ( B X. B )
3		relss 4819	. . . 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 110	. . . . 5 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> x ( R i^i ( B X. B ) ) y )
7		brinxp2 4799	. . . . 5 |- ( x ( R i^i ( B X. B ) ) y <-> ( x e. B /\ y e. B /\ x R y ) )
8	6, 7	sylib 122	. . . 4 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> ( x e. B /\ y e. B /\ x R y ) )
9	8	simp2d 1037	. . 3 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> y e. B )
10	8	simp1d 1036	. . 3 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> x e. B )
11		erinxp.r	. . . . 5 |- ( ph -> R Er A )
12	11	adantr 276	. . . 4 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> R Er A )
13	8	simp3d 1038	. . . 4 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> x R y )
14	12, 13	ersym 6757	. . 3 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> y R x )
15		brinxp2 4799	. . 3 |- ( y ( R i^i ( B X. B ) ) x <-> ( y e. B /\ x e. B /\ y R x ) )
16	9, 10, 14, 15	syl3anbrc 1208	. 2 |- ( ( ph /\ x ( R i^i ( B X. B ) ) y ) -> y ( R i^i ( B X. B ) ) x )
17	10	adantrr 479	. . 3 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x e. B )
18		simprr 533	. . . . 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 4799	. . . . 5 |- ( y ( R i^i ( B X. B ) ) z <-> ( y e. B /\ z e. B /\ y R z ) )
20	18, 19	sylib 122	. . . 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 1037	. . 3 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> z e. B )
22	11	adantr 276	. . . 4 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> R Er A )
23	13	adantrr 479	. . . 4 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x R y )
24	20	simp3d 1038	. . . 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 6761	. . 3 |- ( ( ph /\ ( x ( R i^i ( B X. B ) ) y /\ y ( R i^i ( B X. B ) ) z ) ) -> x R z )
26		brinxp2 4799	. . 3 |- ( x ( R i^i ( B X. B ) ) z <-> ( x e. B /\ z e. B /\ x R z ) )
27	17, 21, 25, 26	syl3anbrc 1208	. 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 276	. . . . . 6 |- ( ( ph /\ x e. B ) -> R Er A )
29		erinxp.a	. . . . . . 7 |- ( ph -> B C_ A )
30	29	sselda 3228	. . . . . 6 |- ( ( ph /\ x e. B ) -> x e. A )
31	28, 30	erref 6765	. . . . 5 |- ( ( ph /\ x e. B ) -> x R x )
32	31	ex 115	. . . 4 |- ( ph -> ( x e. B -> x R x ) )
33	32	pm4.71rd 394	. . 3 |- ( ph -> ( x e. B <-> ( x R x /\ x e. B ) ) )
34		brin 4146	. . . 4 |- ( x ( R i^i ( B X. B ) ) x <-> ( x R x /\ x ( B X. B ) x ) )
35		brxp 4762	. . . . . 6 |- ( x ( B X. B ) x <-> ( x e. B /\ x e. B ) )
36		anidm 396	. . . . . 6 |- ( ( x e. B /\ x e. B ) <-> x e. B )
37	35, 36	bitri 184	. . . . 5 |- ( x ( B X. B ) x <-> x e. B )
38	37	anbi2i 457	. . . 4 |- ( ( x R x /\ x ( B X. B ) x ) <-> ( x R x /\ x e. B ) )
39	34, 38	bitri 184	. . 3 |- ( x ( R i^i ( B X. B ) ) x <-> ( x R x /\ x e. B ) )
40	33, 39	bitr4di 198	. 2 |- ( ph -> ( x e. B <-> x ( R i^i ( B X. B ) ) x ) )
41	5, 16, 27, 40	iserd 6771	1 |- ( ph -> ( R i^i ( B X. B ) ) Er B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   e. wcel 2202   i^i cin 3200   C_ wss 3201   class class class wbr 4093   X. cxp 4729   Rel wrel 4736   Er wer 6742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-er 6745

This theorem is referenced by: (None)