Theorem releldmqscoss 38661

Description: Elementhood in the domain quotient of the class of cosets by a relation. (Contributed by Peter Mazsa, 23-Apr-2021.)

Assertion

Ref	Expression
releldmqscoss	⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅)))

Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝑅,𝑥

Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem releldmqscoss

Step	Hyp	Ref	Expression
1		eldmqs1cossres 38660	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅)))
2	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅)))
3		resdm 6044	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
4	3	cosseqd 38429	. . . . . 6 ⊢ (Rel 𝑅 → ≀ (𝑅 ↾ dom 𝑅) = ≀ 𝑅)
5	4	dmqseqd 38643	. . . . 5 ⊢ (Rel 𝑅 → (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) = (dom ≀ 𝑅 / ≀ 𝑅))
6	5	eleq2d 2827	. . . 4 ⊢ (Rel 𝑅 → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅)))
7	6	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅)))
8	4	eceq2d 8788	. . . . . 6 ⊢ (Rel 𝑅 → [𝑥] ≀ (𝑅 ↾ dom 𝑅) = [𝑥] ≀ 𝑅)
9	8	eqeq2d 2748	. . . . 5 ⊢ (Rel 𝑅 → (𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ 𝐴 = [𝑥] ≀ 𝑅))
10	9	2rexbidv 3222	. . . 4 ⊢ (Rel 𝑅 → (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
11	10	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
12	2, 7, 11	3bitr3d 309	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
13	12	ex 412	1 ⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  dom cdm 5685   ↾ cres 5687  Rel wrel 5690  [cec 8743   / cqs 8744   ≀ ccoss 38182

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-10 2141  ax-11 2157  ax-12 2177  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-nf 1784  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-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751  df-coss 38412

This theorem is referenced by:  disjdmqsss  38803  disjdmqscossss  38804