Theorem erinxp 6116

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 3152	. . . 4 |- ( R i^i (B X. B)) C_ (B X. B)
2		relxp 4390	. . . 4 |- Rel (B X. B)
3		relss 4370	. . . 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 103	. . . . 5 |- ((ph /\ x( R i^i (B X. B))y) -> x( R i^i (B X. B))y)
7		brinxp2 4350	. . . . 5 |- (x( R i^i (B X. B))y <-> (x e. B /\ y e. B /\ x Ry))
8	6, 7	sylib 127	. . . 4 |- ((ph /\ x( R i^i (B X. B))y) -> (x e. B /\ y e. B /\ x Ry))
9	8	simp2d 916	. . 3 |- ((ph /\ x( R i^i (B X. B))y) -> y e. B)
10	8	simp1d 915	. . 3 |- ((ph /\ x( R i^i (B X. B))y) -> x e. B)
11		erinxp.r	. . . . 5 |- (ph -> R Er A)
12	11	adantr 261	. . . 4 |- ((ph /\ x( R i^i (B X. B))y) -> R Er A)
13	8	simp3d 917	. . . 4 |- ((ph /\ x( R i^i (B X. B))y) -> x Ry)
14	12, 13	ersym 6054	. . 3 |- ((ph /\ x( R i^i (B X. B))y) -> y Rx)
15		brinxp2 4350	. . 3 |- (y( R i^i (B X. B))x <-> (y e. B /\ x e. B /\ y Rx))
16	9, 10, 14, 15	syl3anbrc 1087	. 2 |- ((ph /\ x( R i^i (B X. B))y) -> y( R i^i (B X. B))x)
17	10	adantrr 448	. . 3 |- ((ph /\ (x( R i^i (B X. B))y /\ y( R i^i (B X. B))z)) -> x e. B)
18		simprr 484	. . . . 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 4350	. . . . 5 |- (y( R i^i (B X. B))z <-> (y e. B /\ z e. B /\ y Rz))
20	18, 19	sylib 127	. . . 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 Rz))
21	20	simp2d 916	. . 3 |- ((ph /\ (x( R i^i (B X. B))y /\ y( R i^i (B X. B))z)) -> z e. B)
22	11	adantr 261	. . . 4 |- ((ph /\ (x( R i^i (B X. B))y /\ y( R i^i (B X. B))z)) -> R Er A)
23	13	adantrr 448	. . . 4 |- ((ph /\ (x( R i^i (B X. B))y /\ y( R i^i (B X. B))z)) -> x Ry)
24	20	simp3d 917	. . . 4 |- ((ph /\ (x( R i^i (B X. B))y /\ y( R i^i (B X. B))z)) -> y Rz)
25	22, 23, 24	ertrd 6058	. . 3 |- ((ph /\ (x( R i^i (B X. B))y /\ y( R i^i (B X. B))z)) -> x Rz)
26		brinxp2 4350	. . 3 |- (x( R i^i (B X. B))z <-> (x e. B /\ z e. B /\ x Rz))
27	17, 21, 25, 26	syl3anbrc 1087	. 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 261	. . . . . 6 |- ((ph /\ x e. B) -> R Er A)
29		erinxp.a	. . . . . . 7 |- (ph -> B C_ A)
30	29	sselda 2939	. . . . . 6 |- ((ph /\ x e. B) -> x e. A)
31	28, 30	erref 6062	. . . . 5 |- ((ph /\ x e. B) -> x Rx)
32	31	ex 108	. . . 4 |- (ph -> (x e. B -> x Rx))
33	32	pm4.71rd 374	. . 3 |- (ph -> (x e. B <-> (x Rx /\ x e. B)))
34		brin 3802	. . . 4 |- (x( R i^i (B X. B))x <-> (x Rx /\ x(B X. B)x))
35		brxp 4318	. . . . . 6 |- (x(B X. B)x <-> (x e. B /\ x e. B))
36		anidm 376	. . . . . 6 |- ((x e. B /\ x e. B) <-> x e. B)
37	35, 36	bitri 173	. . . . 5 |- (x(B X. B)x <-> x e. B)
38	37	anbi2i 430	. . . 4 |- ((x Rx /\ x(B X. B)x) <-> (x Rx /\ x e. B))
39	34, 38	bitri 173	. . 3 |- (x( R i^i (B X. B))x <-> (x Rx /\ x e. B))
40	33, 39	syl6bbr 187	. 2 |- (ph -> (x e. B <-> x( R i^i (B X. B))x))
41	5, 16, 27, 40	iserd 6068	1 |- (ph -> ( R i^i (B X. B)) Er B)

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 884   e. wcel 1390   i^i cin 2910   C_ wss 2911   class class class wbr 3755   X. cxp 4286   Rel wrel 4293   Er wer 6039

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042

This theorem is referenced by: (None)