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Theorem disjdmqsss 38784
Description: Lemma for disjdmqseq 38787 via disjdmqs 38786. (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 38712 . . . . . 6 ( Disj 𝑅 → Rel 𝑅)
2 releldmqs 38640 . . . . . . 7 (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅)))
32elv 3483 . . . . . 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 38783 . . . . . . . 8 (𝑥 ∈ V → ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅)))
65elv 3483 . . . . . . 7 ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅))
76ralrimivv 3198 . . . . . 6 ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅)
8 2r19.29 3137 . . . . . . 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 2758 . . . . . . 7 ((𝑣 = [𝑢]𝑅 ∧ [𝑢]𝑅 = [𝑥] ≀ 𝑅) → 𝑣 = [𝑥] ≀ 𝑅)
1312ancoms 458 . . . . . 6 (([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅) → 𝑣 = [𝑥] ≀ 𝑅)
1413reximi 3082 . . . . 5 (∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅) → ∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)
1514reximi 3082 . . . 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 38642 . . . . 5 (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)))
1817elv 3483 . . . 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 4001 1 ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 /𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  dom cdm 5689  Rel wrel 5694  [cec 8742   / cqs 8743  ccoss 38162   Disj wdisjALTV 38196
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-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  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-id 5583  df-xp 5695  df-rel 5696  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  df-cnvrefrel 38509  df-disjALTV 38687
This theorem is referenced by:  disjdmqs  38786
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