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	⊢ (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 3430	. . . 4 ⊢ (𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2		relxp 4841	. . . 4 ⊢ Rel (𝐵 × 𝐵)
3		relss 4819	. . . 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 4799	. . . . 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 1037	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦 ∈ 𝐵)
10	8	simp1d 1036	. . 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 1038	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
14	12, 13	ersym 6757	. . 3 ⊢ ((𝜑 ∧ 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
15		brinxp2 4799	. . 3 ⊢ (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦𝑅𝑥))
16	9, 10, 14, 15	syl3anbrc 1208	. 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 4799	. . . . 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 1037	. . 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 1038	. . . 4 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
25	22, 23, 24	ertrd 6761	. . 3 ⊢ ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ∧ 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
26		brinxp2 4799	. . 3 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑥𝑅𝑧))
27	17, 21, 25, 26	syl3anbrc 1208	. 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 3228	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
31	28, 30	erref 6765	. . . . 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 4146	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥 ∧ 𝑥(𝐵 × 𝐵)𝑥))
35		brxp 4762	. . . . . 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 6771	1 ⊢ (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   ∈ wcel 2202   ∩ cin 3200   ⊆ wss 3201   class class class wbr 4093   × 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)