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Theorem releldmqscoss 38642
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 (𝐴𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅)))
Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝑅,𝑥
Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem releldmqscoss
StepHypRef Expression
1 eldmqs1cossres 38641 . . . 4 (𝐴𝑉 → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅)))
21adantr 480 . . 3 ((𝐴𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅)))
3 resdm 6046 . . . . . . 7 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
43cosseqd 38410 . . . . . 6 (Rel 𝑅 → ≀ (𝑅 ↾ dom 𝑅) = ≀ 𝑅)
54dmqseqd 38624 . . . . 5 (Rel 𝑅 → (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) = (dom ≀ 𝑅 /𝑅))
65eleq2d 2825 . . . 4 (Rel 𝑅 → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom ≀ 𝑅 /𝑅)))
76adantl 481 . . 3 ((𝐴𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom ≀ 𝑅 /𝑅)))
84eceq2d 8787 . . . . . 6 (Rel 𝑅 → [𝑥] ≀ (𝑅 ↾ dom 𝑅) = [𝑥] ≀ 𝑅)
98eqeq2d 2746 . . . . 5 (Rel 𝑅 → (𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ 𝐴 = [𝑥] ≀ 𝑅))
1092rexbidv 3220 . . . 4 (Rel 𝑅 → (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
1110adantl 481 . . 3 ((𝐴𝑉 ∧ Rel 𝑅) → (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
122, 7, 113bitr3d 309 . 2 ((𝐴𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
1312ex 412 1 (𝐴𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  dom cdm 5689  cres 5691  Rel wrel 5694  [cec 8742   / cqs 8743  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750  df-coss 38393
This theorem is referenced by:  disjdmqsss  38784  disjdmqscossss  38785
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