Theorem brinxper 8712

Description: Conditions for a reflexive, symmetric and transitive binary relation to be an equivalence relation over a class 𝑉. (Contributed by AV, 11-Jun-2025.)

Hypotheses

Ref	Expression
brinxper.r	⊢ (𝑥 ∈ 𝑉 → 𝑥 ∼ 𝑥)
brinxper.s	⊢ (𝑥 ∈ 𝑉 → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))
brinxper.t	⊢ (𝑥 ∈ 𝑉 → ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧))

Assertion

Ref	Expression
brinxper	⊢ ( ∼ ∩ (𝑉 × 𝑉)) Er 𝑉

Distinct variable groups:   𝑥,𝑉,𝑦,𝑧   𝑥, ∼ ,𝑦,𝑧

Proof of Theorem brinxper

Step	Hyp	Ref	Expression
1		relinxp 5792	. 2 ⊢ Rel ( ∼ ∩ (𝑉 × 𝑉))
2		brxp 5701	. . . . 5 ⊢ (𝑥(𝑉 × 𝑉)𝑦 ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
3		brinxper.s	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))
4	3	adantr 485	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))
5		ancom 465	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉))
6		brxp 5701	. . . . . . 7 ⊢ (𝑦(𝑉 × 𝑉)𝑥 ↔ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉))
7	5, 6	sylbb2 241	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦(𝑉 × 𝑉)𝑥)
8	4, 7	jctird 535	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑥 ∧ 𝑦(𝑉 × 𝑉)𝑥)))
9	2, 8	sylbi 220	. . . 4 ⊢ (𝑥(𝑉 × 𝑉)𝑦 → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑥 ∧ 𝑦(𝑉 × 𝑉)𝑥)))
10	9	impcom 412	. . 3 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) → (𝑦 ∼ 𝑥 ∧ 𝑦(𝑉 × 𝑉)𝑥))
11		brin 5157	. . 3 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 ↔ (𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦))
12		brin 5157	. . 3 ⊢ (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑥 ↔ (𝑦 ∼ 𝑥 ∧ 𝑦(𝑉 × 𝑉)𝑥))
13	10, 11, 12	3imtr4i 295	. 2 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 → 𝑦( ∼ ∩ (𝑉 × 𝑉))𝑥)
14		brin 5157	. . . . . . 7 ⊢ (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧 ↔ (𝑦 ∼ 𝑧 ∧ 𝑦(𝑉 × 𝑉)𝑧))
15		brxp 5701	. . . . . . . . . 10 ⊢ (𝑦(𝑉 × 𝑉)𝑧 ↔ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
16		brinxper.t	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑉 → ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧))
17	16	expd 420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧)))
18	17	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧)))
19	18	impcom 412	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧))
20	19	com12 33	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∼ 𝑧 → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∼ 𝑧))
21	20	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∼ 𝑧))
22	21	imp 411	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) ∧ (𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) → 𝑥 ∼ 𝑧)
23		simplr 780	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) → 𝑧 ∈ 𝑉)
24		simprl 782	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
25	23, 24	anim12ci 625	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) ∧ (𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) → (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
26	22, 25	jca 520	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) ∧ (𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) → (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))
27	26	exp31 424	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 ∼ 𝑧 → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))))
28	15, 27	sylbi 220	. . . . . . . . 9 ⊢ (𝑦(𝑉 × 𝑉)𝑧 → (𝑦 ∼ 𝑧 → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))))
29	28	impcom 412	. . . . . . . 8 ⊢ ((𝑦 ∼ 𝑧 ∧ 𝑦(𝑉 × 𝑉)𝑧) → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))))
30	2	anbi2i 634	. . . . . . . 8 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) ↔ (𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
31		brxp 5701	. . . . . . . . 9 ⊢ (𝑥(𝑉 × 𝑉)𝑧 ↔ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
32	31	anbi2i 634	. . . . . . . 8 ⊢ ((𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧) ↔ (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))
33	29, 30, 32	3imtr4g 299	. . . . . . 7 ⊢ ((𝑦 ∼ 𝑧 ∧ 𝑦(𝑉 × 𝑉)𝑧) → ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧)))
34	14, 33	sylbi 220	. . . . . 6 ⊢ (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧 → ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧)))
35	34	com12 33	. . . . 5 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) → (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧 → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧)))
36	11, 35	sylbi 220	. . . 4 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 → (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧 → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧)))
37	36	imp 411	. . 3 ⊢ ((𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 ∧ 𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧) → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧))
38		brin 5157	. . 3 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑧 ↔ (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧))
39	37, 38	sylibr 237	. 2 ⊢ ((𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 ∧ 𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧) → 𝑥( ∼ ∩ (𝑉 × 𝑉))𝑧)
40		brinxper.r	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∼ 𝑥)
41		id 23	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉)
42		brxp 5701	. . . . . 6 ⊢ (𝑥(𝑉 × 𝑉)𝑥 ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉))
43	41, 41, 42	sylanbrc 594	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → 𝑥(𝑉 × 𝑉)𝑥)
44	40, 43	jca 520	. . . 4 ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∼ 𝑥 ∧ 𝑥(𝑉 × 𝑉)𝑥))
45	42	simplbi 501	. . . . 5 ⊢ (𝑥(𝑉 × 𝑉)𝑥 → 𝑥 ∈ 𝑉)
46	45	adantl 486	. . . 4 ⊢ ((𝑥 ∼ 𝑥 ∧ 𝑥(𝑉 × 𝑉)𝑥) → 𝑥 ∈ 𝑉)
47	44, 46	impbii 212	. . 3 ⊢ (𝑥 ∈ 𝑉 ↔ (𝑥 ∼ 𝑥 ∧ 𝑥(𝑉 × 𝑉)𝑥))
48		brin 5157	. . 3 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑥 ↔ (𝑥 ∼ 𝑥 ∧ 𝑥(𝑉 × 𝑉)𝑥))
49	47, 48	bitr4i 281	. 2 ⊢ (𝑥 ∈ 𝑉 ↔ 𝑥( ∼ ∩ (𝑉 × 𝑉))𝑥)
50	1, 13, 39, 49	iseri 8710	1 ⊢ ( ∼ ∩ (𝑉 × 𝑉)) Er 𝑉

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145   ∩ cin 3906   class class class wbr 5105   × cxp 5650   Er wer 8679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-er 8682

This theorem is referenced by:  gricer  48544  grlicer  48636