Theorem releldmqs 36770

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 5936	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
2	1	dmqseqd 36755	. . . . 5 ⊢ (Rel 𝑅 → (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) = (dom 𝑅 / 𝑅))
3	2	eleq2d 2824	. . . 4 ⊢ (Rel 𝑅 → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom 𝑅 / 𝑅)))
4	3	adantl 482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom 𝑅 / 𝑅)))
5		eldmqsres2 36422	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (dom (𝑅 ↾ dom 𝑅) / (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅))
6	5	adantr 481	. . 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 280	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅))
8	7	ex 413	1 ⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  dom cdm 5589   ↾ cres 5591  Rel wrel 5594  [cec 8496   / cqs 8497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-qs 8504

This theorem is referenced by: (None)