Theorem releldmqs 38829

Description: Elementhood in the domain quotient of a relation. (Contributed by Peter Mazsa, 24-Apr-2021.)

Assertion

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

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

Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem releldmqs

Step	Hyp	Ref	Expression
1		resdm 5982	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
2	1	dmqseqd 38812	. . . . 5 ⊢ (Rel 𝑅 → (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) = (dom 𝑅 / 𝑅))
3	2	eleq2d 2819	. . . 4 ⊢ (Rel 𝑅 → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom 𝑅 / 𝑅)))
4	3	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom 𝑅 / 𝑅)))
5		eldmqsres2 38399	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅))
6	5	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅))
7	4, 6	bitr3d 281	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅))
8	7	ex 412	1 ⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3057  dom cdm 5621   ↾ cres 5623  Rel wrel 5626  [cec 8629   / cqs 8630

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-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8633  df-qs 8637

This theorem is referenced by:  disjdmqsss  38973