Theorem brinxper 8755

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 5812	. 2 ⊢ Rel ( ∼ ∩ (𝑉 × 𝑉))
2		brxp 5723	. . . . 5 ⊢ (𝑥(𝑉 × 𝑉)𝑦 ↔ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
3		brinxper.s	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))
4	3	adantr 479	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))
5		ancom 459	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉))
6		brxp 5723	. . . . . . 7 ⊢ (𝑦(𝑉 × 𝑉)𝑥 ↔ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉))
7	5, 6	sylbb2 237	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦(𝑉 × 𝑉)𝑥)
8	4, 7	jctird 525	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑥 ∧ 𝑦(𝑉 × 𝑉)𝑥)))
9	2, 8	sylbi 216	. . . 4 ⊢ (𝑥(𝑉 × 𝑉)𝑦 → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑥 ∧ 𝑦(𝑉 × 𝑉)𝑥)))
10	9	impcom 406	. . 3 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) → (𝑦 ∼ 𝑥 ∧ 𝑦(𝑉 × 𝑉)𝑥))
11		brin 5197	. . 3 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 ↔ (𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦))
12		brin 5197	. . 3 ⊢ (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑥 ↔ (𝑦 ∼ 𝑥 ∧ 𝑦(𝑉 × 𝑉)𝑥))
13	10, 11, 12	3imtr4i 291	. 2 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 → 𝑦( ∼ ∩ (𝑉 × 𝑉))𝑥)
14		brin 5197	. . . . . . 7 ⊢ (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧 ↔ (𝑦 ∼ 𝑧 ∧ 𝑦(𝑉 × 𝑉)𝑧))
15		brxp 5723	. . . . . . . . . 10 ⊢ (𝑦(𝑉 × 𝑉)𝑧 ↔ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
16		brinxper.t	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑉 → ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧))
17	16	expd 414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧)))
18	17	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧)))
19	18	impcom 406	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧))
20	19	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∼ 𝑧 → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∼ 𝑧))
21	20	adantl 480	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∼ 𝑧))
22	21	imp 405	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) ∧ (𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) → 𝑥 ∼ 𝑧)
23		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) → 𝑧 ∈ 𝑉)
24		simprl 769	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
25	23, 24	anim12ci 612	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) ∧ (𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) → (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
26	22, 25	jca 510	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ 𝑦 ∼ 𝑧) ∧ (𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))) → (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))
27	26	exp31 418	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 ∼ 𝑧 → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))))
28	15, 27	sylbi 216	. . . . . . . . 9 ⊢ (𝑦(𝑉 × 𝑉)𝑧 → (𝑦 ∼ 𝑧 → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))))
29	28	impcom 406	. . . . . . . 8 ⊢ ((𝑦 ∼ 𝑧 ∧ 𝑦(𝑉 × 𝑉)𝑧) → ((𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))))
30	2	anbi2i 621	. . . . . . . 8 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) ↔ (𝑥 ∼ 𝑦 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
31		brxp 5723	. . . . . . . . 9 ⊢ (𝑥(𝑉 × 𝑉)𝑧 ↔ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
32	31	anbi2i 621	. . . . . . . 8 ⊢ ((𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧) ↔ (𝑥 ∼ 𝑧 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))
33	29, 30, 32	3imtr4g 295	. . . . . . 7 ⊢ ((𝑦 ∼ 𝑧 ∧ 𝑦(𝑉 × 𝑉)𝑧) → ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧)))
34	14, 33	sylbi 216	. . . . . 6 ⊢ (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧 → ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧)))
35	34	com12 32	. . . . 5 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑥(𝑉 × 𝑉)𝑦) → (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧 → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧)))
36	11, 35	sylbi 216	. . . 4 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 → (𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧 → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧)))
37	36	imp 405	. . 3 ⊢ ((𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 ∧ 𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧) → (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧))
38		brin 5197	. . 3 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑧 ↔ (𝑥 ∼ 𝑧 ∧ 𝑥(𝑉 × 𝑉)𝑧))
39	37, 38	sylibr 233	. 2 ⊢ ((𝑥( ∼ ∩ (𝑉 × 𝑉))𝑦 ∧ 𝑦( ∼ ∩ (𝑉 × 𝑉))𝑧) → 𝑥( ∼ ∩ (𝑉 × 𝑉))𝑧)
40		brinxper.r	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∼ 𝑥)
41		id 22	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉)
42		brxp 5723	. . . . . 6 ⊢ (𝑥(𝑉 × 𝑉)𝑥 ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉))
43	41, 41, 42	sylanbrc 581	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → 𝑥(𝑉 × 𝑉)𝑥)
44	40, 43	jca 510	. . . 4 ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∼ 𝑥 ∧ 𝑥(𝑉 × 𝑉)𝑥))
45	42	simplbi 496	. . . . 5 ⊢ (𝑥(𝑉 × 𝑉)𝑥 → 𝑥 ∈ 𝑉)
46	45	adantl 480	. . . 4 ⊢ ((𝑥 ∼ 𝑥 ∧ 𝑥(𝑉 × 𝑉)𝑥) → 𝑥 ∈ 𝑉)
47	44, 46	impbii 208	. . 3 ⊢ (𝑥 ∈ 𝑉 ↔ (𝑥 ∼ 𝑥 ∧ 𝑥(𝑉 × 𝑉)𝑥))
48		brin 5197	. . 3 ⊢ (𝑥( ∼ ∩ (𝑉 × 𝑉))𝑥 ↔ (𝑥 ∼ 𝑥 ∧ 𝑥(𝑉 × 𝑉)𝑥))
49	47, 48	bitr4i 277	. 2 ⊢ (𝑥 ∈ 𝑉 ↔ 𝑥( ∼ ∩ (𝑉 × 𝑉))𝑥)
50	1, 13, 39, 49	iseri 8753	1 ⊢ ( ∼ ∩ (𝑉 × 𝑉)) Er 𝑉

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2099   ∩ cin 3945   class class class wbr 5145   × cxp 5672   Er wer 8723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-er 8726

This theorem is referenced by:  grlicer  47542