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Theorem eldmqs1cossres 38641
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 8807 . . 3 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴)))
2 df-rex 3069 . . . 4 (∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)))
3 eldm1cossres2 38443 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅))
43elv 3483 . . . . . 6 (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅)
54anbi1i 624 . . . . 5 ((𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
65exbii 1845 . . . 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 3069 . . . 4 (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
109rexbii 3092 . . 3 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
11 rexcom4 3286 . . . 4 (∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
12 r19.41v 3187 . . . . 5 (∃𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1312exbii 1845 . . . 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 1537  wex 1776  wcel 2106  wrex 3068  Vcvv 3478  dom cdm 5689  cres 5691  [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-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:  releldmqscoss  38642
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