Theorem disjdmqsss 37976

Description: Lemma for disjdmqseq 37979 via disjdmqs 37978. (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.

Step	Hyp	Ref	Expression
1		disjrel 37904	. . . . . 6 ⊢ ( Disj 𝑅 → Rel 𝑅)
2		releldmqs 37832	. . . . . . 7 ⊢ (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅)))
3	2	elv 3479	. . . . . 6 ⊢ (Rel 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅))
4	1, 3	syl 17	. . . . 5 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅))
5		disjlem19 37975	. . . . . . . 8 ⊢ (𝑥 ∈ V → ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅)))
6	5	elv 3479	. . . . . . 7 ⊢ ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅))
7	6	ralrimivv 3197	. . . . . 6 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅)
8		2r19.29 3138	. . . . . . 7 ⊢ ((∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑢]𝑅))
9	8	ex 412	. . . . . 6 ⊢ (∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 → (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅 → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑢]𝑅)))
10	7, 9	syl 17	. . . . 5 ⊢ ( Disj 𝑅 → (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑢]𝑅 → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑢]𝑅)))
11	4, 10	sylbid 239	. . . 4 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑢]𝑅)))
12		eqtr 2754	. . . . . . 7 ⊢ ((𝑣 = [𝑢]𝑅 ∧ [𝑢]𝑅 = [𝑥] ≀ 𝑅) → 𝑣 = [𝑥] ≀ 𝑅)
13	12	ancoms 458	. . . . . 6 ⊢ (([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑢]𝑅) → 𝑣 = [𝑥] ≀ 𝑅)
14	13	reximi 3083	. . . . 5 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑢]𝑅) → ∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)
15	14	reximi 3083	. . . 4 ⊢ (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑢]𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)
16	11, 15	syl6 35	. . 3 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
17		releldmqscoss 37834	. . . . 5 ⊢ (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)))
18	17	elv 3479	. . . 4 ⊢ (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
19	1, 18	syl 17	. . 3 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
20	16, 19	sylibrd 259	. 2 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) → 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)))
21	20	ssrdv 3988	1 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  Vcvv 3473   ⊆ wss 3948  dom cdm 5676  Rel wrel 5681  [cec 8705   / cqs 8706   ≀ ccoss 37347   Disj wdisjALTV 37381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8709  df-qs 8713  df-coss 37585  df-cnvrefrel 37701  df-disjALTV 37879

This theorem is referenced by:  disjdmqs  37978