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Theorem disjdmqsss 37477
Description: Lemma for disjdmqseq 37480 via disjdmqs 37479. (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 37405 . . . . . 6 ( Disj 𝑅 → Rel 𝑅)
2 releldmqs 37333 . . . . . . 7 (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅)))
32elv 3479 . . . . . 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 37476 . . . . . . . 8 (𝑥 ∈ V → ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅)))
65elv 3479 . . . . . . 7 ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅))
76ralrimivv 3197 . . . . . 6 ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅)
8 2r19.29 3138 . . . . . . 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 413 . . . . . 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 239 . . . 4 ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) → ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅)))
12 eqtr 2754 . . . . . . 7 ((𝑣 = [𝑢]𝑅 ∧ [𝑢]𝑅 = [𝑥] ≀ 𝑅) → 𝑣 = [𝑥] ≀ 𝑅)
1312ancoms 459 . . . . . 6 (([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅) → 𝑣 = [𝑥] ≀ 𝑅)
1413reximi 3083 . . . . 5 (∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅𝑣 = [𝑢]𝑅) → ∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)
1514reximi 3083 . . . 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 37335 . . . . 5 (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)))
1817elv 3479 . . . 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 258 . 2 ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) → 𝑣 ∈ (dom ≀ 𝑅 /𝑅)))
2120ssrdv 3984 1 ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 /𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  wrex 3069  Vcvv 3473  wss 3944  dom cdm 5669  Rel wrel 5674  [cec 8684   / cqs 8685  ccoss 36848   Disj wdisjALTV 36882
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rmo 3375  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8688  df-qs 8692  df-coss 37086  df-cnvrefrel 37202  df-disjALTV 37380
This theorem is referenced by:  disjdmqs  37479
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