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.

Step	Hyp	Ref	Expression
1		disjrel 37405	. . . . . 6 ⊢ ( Disj 𝑅 → Rel 𝑅)
2		releldmqs 37333	. . . . . . 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 37476	. . . . . . . 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 413	. . . . . 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 459	. . . . . 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 37335	. . . . 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 258	. 2 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom 𝑅 / 𝑅) → 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)))
21	20	ssrdv 3984	1 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅))

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