Theorem erinxp 6180

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	⊢ (𝜑 → 𝑅 Er 𝐴)
erinxp.a	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
erinxp	⊢ (𝜑 → (𝑅 ∩ (𝐵 × 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 3158	. . . 4 ⊢ (𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2		relxp 4447	. . . 4 ⊢ Rel (𝐵 × 𝐵)
3		relss 4427	. . . 4 ⊢ ((𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵) → (Rel (𝐵 × 𝐵) → Rel (𝑅 ∩ (𝐵 × 𝐵))))
4	1, 2, 3	mp2 16	. . 3 ⊢ Rel (𝑅 ∩ (𝐵 × 𝐵))
5	4	a1i 9	. 2 ⊢ (𝜑 → Rel (𝑅 ∩ (𝐵 × 𝐵)))
6		simpr 103	. . . . 5 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
7		brinxp2 4407	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦))
8	6, 7	sylib 127	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦))
9	8	simp2d 917	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦 ∈ 𝐵)
10	8	simp1d 916	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥 ∈ 𝐵)
11		erinxp.r	. . . . 5 ⊢ (𝜑 → 𝑅 Er 𝐴)
12	11	adantr 261	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
13	8	simp3d 918	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
14	12, 13	ersym 6118	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
15		brinxp2 4407	. . 3 ⊢ (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦𝑅𝑥))
16	9, 10, 14, 15	syl3anbrc 1088	. 2 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
17	10	adantrr 448	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥 ∈ 𝐵)
18		simprr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
19		brinxp2 4407	. . . . 5 ⊢ (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑦𝑅𝑧))
20	18, 19	sylib 127	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑦𝑅𝑧))
21	20	simp2d 917	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧 ∈ 𝐵)
22	11	adantr 261	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
23	13	adantrr 448	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
24	20	simp3d 918	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
25	22, 23, 24	ertrd 6122	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
26		brinxp2 4407	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑥𝑅𝑧))
27	17, 21, 25, 26	syl3anbrc 1088	. 2 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
28	11	adantr 261	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 Er 𝐴)
29		erinxp.a	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
30	29	sselda 2945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
31	28, 30	erref 6126	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥𝑅𝑥)
32	31	ex 108	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥𝑅𝑥))
33	32	pm4.71rd 374	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵)))
34		brin 3811	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥 ∧ 𝑥(𝐵 × 𝐵)𝑥))
35		brxp 4375	. . . . . 6 ⊢ (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵))
36		anidm 376	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐵)
37	35, 36	bitri 173	. . . . 5 ⊢ (𝑥(𝐵 × 𝐵)𝑥 ↔ 𝑥 ∈ 𝐵)
38	37	anbi2i 430	. . . 4 ⊢ ((𝑥𝑅𝑥 ∧ 𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵))
39	34, 38	bitri 173	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵))
40	33, 39	syl6bbr 187	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥))
41	5, 16, 27, 40	iserd 6132	1 ⊢ (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   ∈ wcel 1393   ∩ cin 2916   ⊆ wss 2917   class class class wbr 3764   × cxp 4343  Rel wrel 4350   Er wer 6103

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-er 6106

This theorem is referenced by: (None)