Theorem erinxp 6778

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 3428	. . . 4 ⊢ (𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2		relxp 4835	. . . 4 ⊢ Rel (𝐵 × 𝐵)
3		relss 4813	. . . 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 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
7		brinxp2 4793	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦))
8	6, 7	sylib 122	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦))
9	8	simp2d 1036	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦 ∈ 𝐵)
10	8	simp1d 1035	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥 ∈ 𝐵)
11		erinxp.r	. . . . 5 ⊢ (𝜑 → 𝑅 Er 𝐴)
12	11	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
13	8	simp3d 1037	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
14	12, 13	ersym 6714	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
15		brinxp2 4793	. . 3 ⊢ (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦𝑅𝑥))
16	9, 10, 14, 15	syl3anbrc 1207	. 2 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
17	10	adantrr 479	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥 ∈ 𝐵)
18		simprr 533	. . . . 5 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
19		brinxp2 4793	. . . . 5 ⊢ (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑦𝑅𝑧))
20	18, 19	sylib 122	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑦𝑅𝑧))
21	20	simp2d 1036	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧 ∈ 𝐵)
22	11	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
23	13	adantrr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
24	20	simp3d 1037	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
25	22, 23, 24	ertrd 6718	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
26		brinxp2 4793	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑥𝑅𝑧))
27	17, 21, 25, 26	syl3anbrc 1207	. 2 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
28	11	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 Er 𝐴)
29		erinxp.a	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
30	29	sselda 3227	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
31	28, 30	erref 6722	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥𝑅𝑥)
32	31	ex 115	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥𝑅𝑥))
33	32	pm4.71rd 394	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵)))
34		brin 4141	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥 ∧ 𝑥(𝐵 × 𝐵)𝑥))
35		brxp 4756	. . . . . 6 ⊢ (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵))
36		anidm 396	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐵)
37	35, 36	bitri 184	. . . . 5 ⊢ (𝑥(𝐵 × 𝐵)𝑥 ↔ 𝑥 ∈ 𝐵)
38	37	anbi2i 457	. . . 4 ⊢ ((𝑥𝑅𝑥 ∧ 𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵))
39	34, 38	bitri 184	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥 ∧ 𝑥 ∈ 𝐵))
40	33, 39	bitr4di 198	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥))
41	5, 16, 27, 40	iserd 6728	1 ⊢ (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   ∈ wcel 2202   ∩ cin 3199   ⊆ wss 3200   class class class wbr 4088   × cxp 4723  Rel wrel 4730   Er wer 6699

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-er 6702

This theorem is referenced by: (None)