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Theorem releldmqscoss 37319
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 37318 . . . 4 (𝐴𝑉 → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅)))
21adantr 481 . . 3 ((𝐴𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅)))
3 resdm 6015 . . . . . . 7 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
43cosseqd 37087 . . . . . 6 (Rel 𝑅 → ≀ (𝑅 ↾ dom 𝑅) = ≀ 𝑅)
54dmqseqd 37301 . . . . 5 (Rel 𝑅 → (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) = (dom ≀ 𝑅 /𝑅))
65eleq2d 2818 . . . 4 (Rel 𝑅 → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom ≀ 𝑅 /𝑅)))
76adantl 482 . . 3 ((𝐴𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ (𝑅 ↾ dom 𝑅) / ≀ (𝑅 ↾ dom 𝑅)) ↔ 𝐴 ∈ (dom ≀ 𝑅 /𝑅)))
84eceq2d 8725 . . . . . 6 (Rel 𝑅 → [𝑥] ≀ (𝑅 ↾ dom 𝑅) = [𝑥] ≀ 𝑅)
98eqeq2d 2742 . . . . 5 (Rel 𝑅 → (𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ 𝐴 = [𝑥] ≀ 𝑅))
1092rexbidv 3218 . . . 4 (Rel 𝑅 → (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
1110adantl 482 . . 3 ((𝐴𝑉 ∧ Rel 𝑅) → (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ (𝑅 ↾ dom 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
122, 7, 113bitr3d 308 . 2 ((𝐴𝑉 ∧ Rel 𝑅) → (𝐴 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))
1312ex 413 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 205  wa 396   = wceq 1541  wcel 2106  wrex 3069  dom cdm 5666  cres 5668  Rel wrel 5671  [cec 8681   / cqs 8682  ccoss 36832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8685  df-qs 8689  df-coss 37070
This theorem is referenced by:  disjdmqsss  37461  disjdmqscossss  37462
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