Theorem eleqvrels3 38549

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 38545	. 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
2		dmeq 5928	. . . 4 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3		breq 5168	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑥 ↔ 𝑥𝑅𝑥))
4	2, 3	raleqbidv 3354	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
5		breq 5168	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑦 ↔ 𝑥𝑅𝑦))
6		breq 5168	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑥 ↔ 𝑦𝑅𝑥))
7	5, 6	imbi12d 344	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 → 𝑦𝑟𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
8	7	2albidv 1922	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
9		breq 5168	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑦𝑟𝑧 ↔ 𝑦𝑅𝑧))
10	5, 9	anbi12d 631	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
11		breq 5168	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥𝑟𝑧 ↔ 𝑥𝑅𝑧))
12	10, 11	imbi12d 344	. . . . 5 ⊢ (𝑟 = 𝑅 → (((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
13	12	2albidv 1922	. . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
14	13	albidv 1919	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
15	4, 8, 14	3anbi123d 1436	. 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 38210	1 ⊢ (𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∀wral 3067   class class class wbr 5166  dom cdm 5700   Rels crels 38137   EqvRels ceqvrels 38151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-rels 38441  df-ssr 38454  df-refs 38466  df-refrels 38467  df-syms 38498  df-symrels 38499  df-trs 38528  df-trrels 38529  df-eqvrels 38540

This theorem is referenced by: (None)