Theorem bcth3 25316

Description: Baire's Category Theorem, version 3: The intersection of countably many dense open sets is dense. (Contributed by Mario Carneiro, 10-Jan-2014.)

Hypothesis

Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)

Assertion

Ref	Expression
bcth3	⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋) → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋)

Distinct variable groups:   𝐷,𝑘   𝑘,𝐽   𝑘,𝑀   𝑘,𝑋

Proof of Theorem bcth3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmetmet 25271	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2		metxmet 24317	. . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
3	1, 2	syl 17	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4		bcth.2	. . . . . . . . . 10 ⊢ 𝐽 = (MetOpen‘𝐷)
5	4	mopntop 24423	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
6	5	ad2antrr 732	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7		ffvelcdm 7022	. . . . . . . . . 10 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ 𝐽)
8		elssuni 4869	. . . . . . . . . 10 ⊢ ((𝑀‘𝑘) ∈ 𝐽 → (𝑀‘𝑘) ⊆ ∪ 𝐽)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
10	9	adantll 720	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
11		eqid 2739	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
12	11	clsval2 23033	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑘) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
13	6, 10, 12	syl2anc 590	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
14	4	mopnuni 24424	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
15	14	ad2antrr 732	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = ∪ 𝐽)
16	13, 15	eqeq12d 2755	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋 ↔ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽))
17		difeq2 4051	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = (∪ 𝐽 ∖ ∪ 𝐽))
18		difid 4304	. . . . . . . 8 ⊢ (∪ 𝐽 ∖ ∪ 𝐽) = ∅
19	17, 18	eqtrdi 2790	. . . . . . 7 ⊢ ((∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ∅)
20		difss 4066	. . . . . . . . . . . 12 ⊢ (∪ 𝐽 ∖ (𝑀‘𝑘)) ⊆ ∪ 𝐽
21	11	ntropn 23032	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ (𝑀‘𝑘)) ⊆ ∪ 𝐽) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
22	6, 20, 21	sylancl 592	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
23		elssuni 4869	. . . . . . . . . . 11 ⊢ (((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽 → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ⊆ ∪ 𝐽)
24	22, 23	syl 17	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ⊆ ∪ 𝐽)
25		dfss4 4197	. . . . . . . . . 10 ⊢ (((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))
26	24, 25	sylib 219	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))
27		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
28		elfvdm 6861	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
29	28	difexd 5259	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑘)) ∈ V)
30	29	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ (𝑀‘𝑘)) ∈ V)
31		fveq2 6827	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝑀‘𝑥) = (𝑀‘𝑘))
32	31	difeq2d 4057	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑋 ∖ (𝑀‘𝑥)) = (𝑋 ∖ (𝑀‘𝑘)))
33		eqid 2739	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))
34	32, 33	fvmptg 6933	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀‘𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (𝑋 ∖ (𝑀‘𝑘)))
35	27, 30, 34	syl2anr 603	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (𝑋 ∖ (𝑀‘𝑘)))
36	15	difeq1d 4056	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑘)) = (∪ 𝐽 ∖ (𝑀‘𝑘)))
37	35, 36	eqtrd 2774	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (∪ 𝐽 ∖ (𝑀‘𝑘)))
38	37	fveq2d 6831	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))
39	26, 38	eqtr4d 2777	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)))
40	39	eqeq1d 2741	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅))
41	19, 40	imbitrid 245	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅))
42	16, 41	sylbid 241	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋 → ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅))
43	42	ralimdva 3151	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅))
44	3, 43	sylan 586	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅))
45		ffvelcdm 7022	. . . . . . . . 9 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑥 ∈ ℕ) → (𝑀‘𝑥) ∈ 𝐽)
46	14	difeq1d 4056	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑥)) = (∪ 𝐽 ∖ (𝑀‘𝑥)))
47	46	adantr 481	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀‘𝑥)) = (∪ 𝐽 ∖ (𝑀‘𝑥)))
48	11	opncld 23016	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
49	5, 48	sylan 586	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
50	47, 49	eqeltrd 2839	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
51	45, 50	sylan2 599	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀:ℕ⟶𝐽 ∧ 𝑥 ∈ ℕ)) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
52	51	anassrs 468	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
53	52	ralrimiva 3131	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
54	3, 53	sylan 586	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
55	33	fmpt 7051	. . . . 5 ⊢ (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽) ↔ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽))
56	54, 55	sylib 219	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽))
57		nne 2938	. . . . . . 7 ⊢ (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
58	57	ralbii 3085	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
59		ralnex 3065	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅)
60	58, 59	bitr3i 278	. . . . 5 ⊢ (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅)
61	4	bcth 25314	. . . . . . 7 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) ≠ ∅) → ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅)
62	61	3expia 1127	. . . . . 6 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽)) → (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) ≠ ∅ → ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅))
63	62	necon1bd 2952	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽)) → (¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ → ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅))
64	60, 63	biimtrid 243	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽)) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅ → ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅))
65	56, 64	syldan 597	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅ → ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅))
66		difeq2 4051	. . . . 5 ⊢ (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅ → (∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = (∪ 𝐽 ∖ ∅))
67	28	difexd 5259	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
68	67	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
69	68	ralrimiva 3131	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
70	33	fnmpt 6625	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) Fn ℕ)
71		fniunfv 7191	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) Fn ℕ → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))
72	69, 70, 71	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))
73	35	iuneq2dv 4946	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘)))
74	32	cbviunv 4968	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘))
75	73, 74	eqtr4di 2792	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)))
76	72, 75	eqtr3d 2776	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)))
77		iundif2 5003	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) = (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥))
78	76, 77	eqtrdi 2790	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥)))
79		ffn 6655	. . . . . . . . . . . . 13 ⊢ (𝑀:ℕ⟶𝐽 → 𝑀 Fn ℕ)
80	79	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
81		fniinfv 6905	. . . . . . . . . . . 12 ⊢ (𝑀 Fn ℕ → ∩ 𝑥 ∈ ℕ (𝑀‘𝑥) = ∩ ran 𝑀)
82	80, 81	syl 17	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ 𝑥 ∈ ℕ (𝑀‘𝑥) = ∩ ran 𝑀)
83	82	difeq2d 4057	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥)) = (𝑋 ∖ ∩ ran 𝑀))
84	14	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑋 = ∪ 𝐽)
85	84	difeq1d 4056	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ ∩ ran 𝑀) = (∪ 𝐽 ∖ ∩ ran 𝑀))
86	78, 83, 85	3eqtrd 2778	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (∪ 𝐽 ∖ ∩ ran 𝑀))
87	86	fveq2d 6831	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀)))
88	87	difeq2d 4057	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
89	5	adantr 481	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝐽 ∈ Top)
90		1nn 12176	. . . . . . . . 9 ⊢ 1 ∈ ℕ
91		biidd 263	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽) ↔ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽)))
92		fnfvelrn 7021	. . . . . . . . . . . . . 14 ⊢ ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
93	80, 92	sylan 586	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
94		intss1 4893	. . . . . . . . . . . . 13 ⊢ ((𝑀‘𝑘) ∈ ran 𝑀 → ∩ ran 𝑀 ⊆ (𝑀‘𝑘))
95	93, 94	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ∩ ran 𝑀 ⊆ (𝑀‘𝑘))
96	95, 10	sstrd 3925	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ∩ ran 𝑀 ⊆ ∪ 𝐽)
97	96	expcom 414	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽))
98	91, 97	vtoclga 3520	. . . . . . . . 9 ⊢ (1 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽))
99	90, 98	ax-mp 5	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽)
100	11	clsval2 23033	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ∩ ran 𝑀 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
101	89, 99, 100	syl2anc 590	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
102	88, 101	eqtr4d 2777	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = ((cls‘𝐽)‘∩ ran 𝑀))
103		dif0 4306	. . . . . . 7 ⊢ (∪ 𝐽 ∖ ∅) = ∪ 𝐽
104	103, 84	eqtr4id 2793	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ∅) = 𝑋)
105	102, 104	eqeq12d 2755	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = (∪ 𝐽 ∖ ∅) ↔ ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋))
106	66, 105	imbitrid 245	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅ → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋))
107	3, 106	sylan 586	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅ → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋))
108	44, 65, 107	3syld 60	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋 → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋))
109	108	3impia 1123	1 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋) → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4261  ∪ cuni 4838  ∩ cint 4877  ∪ ciun 4921  ∩ ciin 4922   ↦ cmpt 5153  dom cdm 5618  ran crn 5619   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  1c1 11030  ℕcn 12165  ∞Metcxmet 21332  Metcmet 21333  MetOpencmopn 21337  Topctop 22876  Clsdccld 22999  intcnt 23000  clsccl 23001  CMetccmet 25239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553  ax-dc 10359  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890  df-rp 12934  df-xneg 13054  df-xadd 13055  df-xmul 13056  df-ico 13295  df-rest 17376  df-topgen 17397  df-psmet 21339  df-xmet 21340  df-met 21341  df-bl 21342  df-mopn 21343  df-fbas 21344  df-fg 21345  df-top 22877  df-topon 22894  df-bases 22929  df-cld 23002  df-ntr 23003  df-cls 23004  df-nei 23081  df-lm 23212  df-fil 23829  df-fm 23921  df-flim 23922  df-flf 23923  df-cfil 25240  df-cau 25241  df-cmet 25242

This theorem is referenced by: (None)