Theorem disjdmqscossss 39062

Description: Lemma for disjdmqseq 39064 via disjdmqs 39063. (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 38989	. . . . . . . 8 ⊢ ( Disj 𝑅 → Rel 𝑅)
2		releldmqscoss 38919	. . . . . . . . 9 ⊢ (𝑣 ∈ V → (Rel 𝑅 → (𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝑣 = [𝑥] ≀ 𝑅)))
3	2	elv 3445	. . . . . . . 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 39060	. . . . . . . . . 10 ⊢ (𝑥 ∈ V → ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅)))
6	5	elv 3445	. . . . . . . . 9 ⊢ ( Disj 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑥 ∈ [𝑢]𝑅) → [𝑢]𝑅 = [𝑥] ≀ 𝑅))
7	6	ralrimivv 3177	. . . . . . . 8 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = [𝑥] ≀ 𝑅)
8		2r19.29 3122	. . . . . . . . 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 2758	. . . . . . . 8 ⊢ (([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → [𝑢]𝑅 = 𝑣)
13	12	reximi 3074	. . . . . . 7 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅([𝑢]𝑅 = [𝑥] ≀ 𝑅 ∧ 𝑣 = [𝑥] ≀ 𝑅) → ∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣)
14	13	reximi 3074	. . . . . 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 3061	. . . . . . . 8 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅[𝑢]𝑅 = 𝑣 ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ [𝑢]𝑅 = 𝑣))
17		19.41v 1950	. . . . . . . 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 3074	. . . . 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 2743	. . . . 5 ⊢ ([𝑢]𝑅 = 𝑣 ↔ 𝑣 = [𝑢]𝑅)
23	22	rexbii 3083	. . . 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 4017	. 2 ⊢ ( Disj 𝑅 → {𝑣 ∣ 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)} ⊆ {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅})
26		abid1 2872	. 2 ⊢ (dom ≀ 𝑅 / ≀ 𝑅) = {𝑣 ∣ 𝑣 ∈ (dom ≀ 𝑅 / ≀ 𝑅)}
27		df-qs 8641	. 2 ⊢ (dom 𝑅 / 𝑅) = {𝑣 ∣ ∃𝑢 ∈ dom 𝑅 𝑣 = [𝑢]𝑅}
28	25, 26, 27	3sstr4g 3987	1 ⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  dom cdm 5624  Rel wrel 5629  [cec 8633   / cqs 8634   ≀ ccoss 38383   Disj wdisjALTV 38417

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rmo 3350  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637  df-qs 8641  df-coss 38674  df-cnvrefrel 38780  df-disjALTV 38964

This theorem is referenced by:  disjdmqs  39063