Theorem disjdmqscossss 38795

Description: Lemma for disjdmqseq 38797 via disjdmqs 38796. (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 38722	. . . . . . . 8 ⊢ ( Disj 𝑅 → Rel 𝑅)
2		releldmqscoss 38652	. . . . . . . . 9 ⊢ (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)))
3	2	elv 3452	. . . . . . . 8 ⊢ (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
4	1, 3	syl 17	. . . . . . 7 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅))
5		disjlem19 38793	. . . . . . . . . 10 ⊢ (𝑥 ∈ V → ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅)))
6	5	elv 3452	. . . . . . . . 9 ⊢ ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅))
7	6	ralrimivv 3178	. . . . . . . 8 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅)
8		2r19.29 3119	. . . . . . . . 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 412	. . . . . . . 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 17	. . . . . . 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 240	. . . . . 6 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅)))
12		eqtr3 2751	. . . . . . . 8 ⊢ (([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → [𝑢]𝑅 = 𝑣)
13	12	reximi 3067	. . . . . . 7 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → ∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣)
14	13	reximi 3067	. . . . . 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 35	. . . . 5 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣))
16		df-rex 3054	. . . . . . . 8 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
17		19.41v 1949	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
18	16, 17	bitri 275	. . . . . . 7 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
19	18	simprbi 496	. . . . . 6 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 → [𝑢]𝑅 = 𝑣)
20	19	reximi 3067	. . . . 5 ⊢ (∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 → ∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣)
21	15, 20	syl6 35	. . . 4 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣))
22		eqcom 2736	. . . . 5 ⊢ ([𝑢]𝑅 = 𝑣 ↔ 𝑣 = [𝑢]𝑅)
23	22	rexbii 3076	. . . 4 ⊢ (∃𝑢 ∈ dom 𝑅[𝑢]𝑅 = 𝑣 ↔ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅)
24	21, 23	imbitrdi 251	. . 3 ⊢ ( Disj 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) → ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅))
25	24	ss2abdv 4029	. 2 ⊢ ( Disj 𝑅 → {𝑣 ∣ 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)} ⊆ {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅})
26		abid1 2864	. 2 ⊢ (dom ≀ 𝑅 / ≀ 𝑅) = {𝑣 ∣ 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)}
27		df-qs 8677	. 2 ⊢ (dom 𝑅 / 𝑅) = {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅}
28	25, 26, 27	3sstr4g 4000	1 ⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  dom cdm 5638  Rel wrel 5643  [cec 8669   / cqs 8670   ≀ ccoss 38169   Disj wdisjALTV 38203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rmo 3354  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673  df-qs 8677  df-coss 38402  df-cnvrefrel 38518  df-disjALTV 38697

This theorem is referenced by:  disjdmqs  38796