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Theorem eldmqs1cossres 36069
 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 8333 . . 3 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴)))
2 df-rex 3112 . . . 4 (∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)))
3 eldm1cossres2 35877 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅))
43elv 3446 . . . . . 6 (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅)
54anbi1i 626 . . . . 5 ((𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
65exbii 1849 . . . 4 (∃𝑥(𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
72, 6bitri 278 . . 3 (∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
81, 7syl6bb 290 . 2 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴))))
9 df-rex 3112 . . . 4 (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
109rexbii 3210 . . 3 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
11 rexcom4 3212 . . . 4 (∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
12 r19.41v 3300 . . . . 5 (∃𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1312exbii 1849 . . . 4 (∃𝑥𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1411, 13bitri 278 . . 3 (∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1510, 14bitri 278 . 2 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
168, 15bitr4di 292 1 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441  dom cdm 5519   ↾ cres 5521  [cec 8272   / cqs 8273   ≀ ccoss 35629 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ec 8276  df-qs 8280  df-coss 35835 This theorem is referenced by:  releldmqscoss  36070
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