Theorem eleqvrels2 38593

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

Assertion

Ref	Expression
eleqvrels2	⊢ (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))

Proof of Theorem eleqvrels2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfeqvrels2 38589	. 2 ⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
2		dmeq 5914	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3	2	reseq2d 5997	. . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅))
4		id 22	. . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
5	3, 4	sseq12d 4017	. . 3 ⊢ (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
6		cnveq 5884	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
7	6, 4	sseq12d 4017	. . 3 ⊢ (𝑟 = 𝑅 → (◡𝑟 ⊆ 𝑟 ↔ ◡𝑅 ⊆ 𝑅))
8		coideq 38248	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑅))
9	8, 4	sseq12d 4017	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅))
10	5, 7, 9	3anbi123d 1438	. 2 ⊢ (𝑟 = 𝑅 → ((( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅)))
11	1, 10	rabeqel 38255	1 ⊢ (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951   I cid 5577  ^◡ccnv 5684  dom cdm 5685   ↾ cres 5687   ∘ ccom 5689   Rels crels 38184   EqvRels ceqvrels 38198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-rels 38486  df-ssr 38499  df-refs 38511  df-refrels 38512  df-syms 38543  df-symrels 38544  df-trs 38573  df-trrels 38574  df-eqvrels 38585

This theorem is referenced by:  eleqvrelsrel  38595