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Theorem eldmqs1cossres 38660
Description: Elementhood in the domain quotient of the class of cosets by a restriction. (Contributed by Peter Mazsa, 4-May-2019.)
Assertion
Ref Expression
eldmqs1cossres (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑢,𝑅,𝑥
Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem eldmqs1cossres
StepHypRef Expression
1 elqsg 8808 . . 3 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴)))
2 df-rex 3071 . . . 4 (∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)))
3 eldm1cossres2 38462 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅))
43elv 3485 . . . . . 6 (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅)
54anbi1i 624 . . . . 5 ((𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
65exbii 1848 . . . 4 (∃𝑥(𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
72, 6bitri 275 . . 3 (∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
81, 7bitrdi 287 . 2 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴))))
9 df-rex 3071 . . . 4 (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
109rexbii 3094 . . 3 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
11 rexcom4 3288 . . . 4 (∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
12 r19.41v 3189 . . . . 5 (∃𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1312exbii 1848 . . . 4 (∃𝑥𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1411, 13bitri 275 . . 3 (∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1510, 14bitri 275 . 2 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
168, 15bitr4di 289 1 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  Vcvv 3480  dom cdm 5685  cres 5687  [cec 8743   / cqs 8744  ccoss 38182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751  df-coss 38412
This theorem is referenced by:  releldmqscoss  38661
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