Theorem releldmqscoss 39642

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 39641	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅)))
2	1	adantr 486	. . 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 6017	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
4	3	cosseqd 39415	. . . . . 6 ⊢ (Rel 𝑅 → ≀ (𝑅 ↾ dom 𝑅) = ≀ 𝑅)
5	4	dmqseqd 39623	. . . . 5 ⊢ (Rel 𝑅 → (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) = (dom ≀ 𝑅 / ≀ 𝑅))
6	5	eleq2d 2847	. . . 4 ⊢ (Rel 𝑅 → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅)))
7	6	adantl 487	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅)))
8	4	eceq2d 8745	. . . . . 6 ⊢ (Rel 𝑅 → [𝑥] ≀ (𝑅 ↾ dom 𝑅) = [𝑥] ≀ 𝑅)
9	8	eqeq2d 2772	. . . . 5 ⊢ (Rel 𝑅 → (𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ 𝐴 = [𝑥] ≀ 𝑅))
10	9	2rexbidv 3228	. . . 4 ⊢ (Rel 𝑅 → (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
11	10	adantl 487	. . 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 312	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
13	12	ex 418	1 ⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  dom cdm 5651   ↾ cres 5653  Rel wrel 5656  [cec 8699   / cqs 8700   ≀ ccoss 39080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8703  df-qs 8707  df-coss 39398

This theorem is used by:  disjdmqsss  39802  disjdmqscossss  39803