Theorem eleqvrels3 39216

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 39212	. 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
2		dmeq 5894	. . . 4 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		breq 5115	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑥 ↔ 𝑥𝑅𝑥))
4	2, 3	raleqbidv 3345	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
5		breq 5115	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
6		breq 5115	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑥 ↔ 𝑦𝑅𝑥))
7	5, 6	imbi12d 347	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 → 𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
8	7	2albidv 1950	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
9		breq 5115	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑧 ↔ 𝑦𝑅𝑧))
10	5, 9	anbi12d 643	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
11		breq 5115	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑧 ↔ 𝑥𝑅𝑧))
12	10, 11	imbi12d 347	. . . . 5 ⊢ (𝑟 = 𝑅 → (((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
13	12	2albidv 1950	. . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
14	13	albidv 1947	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
15	4, 8, 14	3anbi123d 1462	. 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 38796	1 ⊢ (𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∀wal 1565   = wceq 1567   ∈ wcel 2149  ∀wral 3085   class class class wbr 5113  dom cdm 5662   Rels crels 38724   EqvRels ceqvrels 38738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-rels 38979  df-ssr 39117  df-refs 39129  df-refrels 39130  df-syms 39161  df-symrels 39162  df-trs 39195  df-trrels 39196  df-eqvrels 39207

This theorem is referenced by: (None)