Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjdmqsss Structured version   Visualization version   GIF version

Theorem disjdmqsss 38803
Description: Lemma for disjdmqseq 38806 via disjdmqs 38805. (Contributed by Peter Mazsa, 16-Sep-2021.)
Assertion
Ref Expression
disjdmqsss ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 /𝑅))

Proof of Theorem disjdmqsss
Dummy variables 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disjrel 38731 . . . . . 6 ( Disj 𝑅 → Rel 𝑅)
2 releldmqs 38659 . . . . . . 7 (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅)))
32elv 3485 . . . . . 6 (Rel 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅))
41, 3syl 17 . . . . 5 ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅))
5 disjlem19 38802 . . . . . . . 8 (𝑥 ∈ V → ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅)))
65elv 3485 . . . . . . 7 ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅))
76ralrimivv 3200 . . . . . 6 ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅)
8 2r19.29 3139 . . . . . . 7 ((∀𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅) → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅))
98ex 412 . . . . . 6 (∀𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 → (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅 → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅)))
107, 9syl 17 . . . . 5 ( Disj 𝑅 → (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅 → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅)))
114, 10sylbid 240 . . . 4 ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅)))
12 eqtr 2760 . . . . . . 7 ((𝑣 = [𝑢]𝑅 ∧ [𝑢]𝑅 = [𝑥] ≀ 𝑅) → 𝑣 = [𝑥] ≀ 𝑅)
1312ancoms 458 . . . . . 6 (([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅) → 𝑣 = [𝑥] ≀ 𝑅)
1413reximi 3084 . . . . 5 (∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅) → ∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)
1514reximi 3084 . . . 4 (∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅) → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)
1611, 15syl6 35 . . 3 ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
17 releldmqscoss 38661 . . . . 5 (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)))
1817elv 3485 . . . 4 (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
191, 18syl 17 . . 3 ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
2016, 19sylibrd 259 . 2 ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) → 𝑣 ∈ (dom ≀ 𝑅 /𝑅)))
2120ssrdv 3989 1 ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 /𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  dom cdm 5685  Rel wrel 5690  [cec 8743   / cqs 8744  ccoss 38182   Disj wdisjALTV 38216
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-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380  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-id 5578  df-xp 5691  df-rel 5692  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  df-cnvrefrel 38528  df-disjALTV 38706
This theorem is referenced by:  disjdmqs  38805
  Copyright terms: Public domain W3C validator