Theorem disjdmqscossss 39615

Description: Lemma for disjdmqseq 39617 via disjdmqs 39616. (Contributed by Peter Mazsa, 16-Sep-2021.)

Assertion

Ref	Expression
disjdmqscossss	⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))

Proof of Theorem disjdmqscossss

Dummy variables 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjrel 39539	. . . . . . . 8 ⊢ ( Disj 𝑅 → Rel 𝑅)
2		releldmqscoss 39454	. . . . . . . . 9 ⊢ (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)))
3	2	elv 3462	. . . . . . . 8 ⊢ (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
4	1, 3	syl 18	. . . . . . 7 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
5		disjlem19 39613	. . . . . . . . . 10 ⊢ (𝑥 ∈ V → ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅)))
6	5	elv 3462	. . . . . . . . 9 ⊢ ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅))
7	6	ralrimivv 3208	. . . . . . . 8 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅)
8		2r19.29 3153	. . . . . . . . 9 ⊢ ((∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅))
9	8	ex 418	. . . . . . . 8 ⊢ (∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅 → (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅 → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅)))
10	7, 9	syl 18	. . . . . . 7 ⊢ ( Disj 𝑅 → (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅 → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅)))
11	4, 10	sylbid 243	. . . . . 6 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅)))
12		eqtr3 2787	. . . . . . . 8 ⊢ (([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → [𝑢]𝑅 = 𝑣)
13	12	reximi 3105	. . . . . . 7 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → ∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣)
14	13	reximi 3105	. . . . . 6 ⊢ (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣)
15	11, 14	syl6 36	. . . . 5 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣))
16		df-rex 3092	. . . . . . . 8 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
17		19.41v 1982	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
18	16, 17	bitri 278	. . . . . . 7 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
19	18	simprbi 503	. . . . . 6 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 → [𝑢]𝑅 = 𝑣)
20	19	reximi 3105	. . . . 5 ⊢ (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 → ∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣)
21	15, 20	syl6 36	. . . 4 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣))
22		eqcom 2772	. . . . 5 ⊢ ([𝑢]𝑅 = 𝑣 ↔ 𝑣 = [𝑢]𝑅)
23	22	rexbii 3114	. . . 4 ⊢ (∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣 ↔ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅)
24	21, 23	imbitrdi 254	. . 3 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅))
25	24	ss2abdv 4020	. 2 ⊢ ( Disj 𝑅 → {𝑣 ∣ 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)} ⊆ {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅})
26		abid1 2901	. 2 ⊢ (dom ≀ 𝑅 / ≀ 𝑅) = {𝑣 ∣ 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)}
27		df-qs 8706	. 2 ⊢ (dom 𝑅 / 𝑅) = {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅}
28	25, 26, 27	3sstr4g 3991	1 ⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  dom cdm 5663  Rel wrel 5668  [cec 8698   / cqs 8699   ≀ ccoss 38892   Disj wdisjALTV 38928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rmo 3371  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8702  df-qs 8706  df-coss 39210  df-cnvrefrel 39316  df-disjALTV 39499

This theorem is used by:  disjdmqs  39616