Theorem releldmqscoss 38778

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 38777	. . . 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 5979	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
4	3	cosseqd 38550	. . . . . 6 ⊢ (Rel 𝑅 → ≀ (𝑅 ↾ dom 𝑅) = ≀ 𝑅)
5	4	dmqseqd 38759	. . . . 5 ⊢ (Rel 𝑅 → (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) = (dom ≀ 𝑅 / ≀ 𝑅))
6	5	eleq2d 2819	. . . 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 8671	. . . . . 6 ⊢ (Rel 𝑅 → [𝑥] ≀ (𝑅 ↾ dom 𝑅) = [𝑥] ≀ 𝑅)
9	8	eqeq2d 2744	. . . . 5 ⊢ (Rel 𝑅 → (𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ 𝐴 = [𝑥] ≀ 𝑅))
10	9	2rexbidv 3198	. . . 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 1541   ∈ wcel 2113  ∃wrex 3057  dom cdm 5619   ↾ cres 5621  Rel wrel 5624  [cec 8626   / cqs 8627   ≀ ccoss 38242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ec 8630  df-qs 8634  df-coss 38533

This theorem is referenced by:  disjdmqsss  38920  disjdmqscossss  38921