Theorem eleqvrels3 39044

Description: Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.)

Assertion

Ref	Expression
eleqvrels3	⊢ (𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels ))

Distinct variable group:   𝑥,𝑅,𝑦,𝑧

Proof of Theorem eleqvrels3

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfeqvrels3 39040	. 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
2		dmeq 5845	. . . 4 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		breq 5074	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑥 ↔ 𝑥𝑅𝑥))
4	2, 3	raleqbidv 3313	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
5		breq 5074	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
6		breq 5074	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑥 ↔ 𝑦𝑅𝑥))
7	5, 6	imbi12d 345	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 → 𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
8	7	2albidv 1930	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
9		breq 5074	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑧 ↔ 𝑦𝑅𝑧))
10	5, 9	anbi12d 638	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
11		breq 5074	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑧 ↔ 𝑥𝑅𝑧))
12	10, 11	imbi12d 345	. . . . 5 ⊢ (𝑟 = 𝑅 → (((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
13	12	2albidv 1930	. . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
14	13	albidv 1927	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
15	4, 8, 14	3anbi123d 1444	. 2 ⊢ (𝑟 = 𝑅 → ((∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
16	1, 15	rabeqel 38624	1 ⊢ (𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels ))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092  ∀wal 1545   = wceq 1547   ∈ wcel 2119  ∀wral 3053   class class class wbr 5072  dom cdm 5618   Rels crels 38552   EqvRels ceqvrels 38566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-rels 38807  df-ssr 38945  df-refs 38957  df-refrels 38958  df-syms 38989  df-symrels 38990  df-trs 39023  df-trrels 39024  df-eqvrels 39035

This theorem is referenced by: (None)