Theorem bcth3 25458

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 25413	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2		metxmet 24459	. . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
3	1, 2	syl 18	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4		bcth.2	. . . . . . . . . 10 ⊢ 𝐽 = (MetOpen‘𝐷)
5	4	mopntop 24565	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
6	5	ad2antrr 738	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7		ffvelcdm 7077	. . . . . . . . . 10 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ 𝐽)
8		elssuni 4908	. . . . . . . . . 10 ⊢ ((𝑀‘𝑘) ∈ 𝐽 → (𝑀‘𝑘) ⊆ ∪ 𝐽)
9	7, 8	syl 18	. . . . . . . . 9 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
10	9	adantll 726	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
11		eqid 2769	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
12	11	clsval2 23175	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑘) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
13	6, 10, 12	syl2anc 595	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
14	4	mopnuni 24566	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
15	14	ad2antrr 738	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = ∪ 𝐽)
16	13, 15	eqeq12d 2785	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋 ↔ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽))
17		difeq2 4083	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = (∪ 𝐽 ∖ ∪ 𝐽))
18		difid 4339	. . . . . . . 8 ⊢ (∪ 𝐽 ∖ ∪ 𝐽) = ∅
19	17, 18	eqtrdi 2820	. . . . . . 7 ⊢ ((∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ∅)
20		difss 4098	. . . . . . . . . . . 12 ⊢ (∪ 𝐽 ∖ (𝑀‘𝑘)) ⊆ ∪ 𝐽
21	11	ntropn 23174	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ (𝑀‘𝑘)) ⊆ ∪ 𝐽) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
22	6, 20, 21	sylancl 597	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
23		elssuni 4908	. . . . . . . . . . 11 ⊢ (((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽 → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ⊆ ∪ 𝐽)
24	22, 23	syl 18	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ⊆ ∪ 𝐽)
25		dfss4 4230	. . . . . . . . . 10 ⊢ (((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))
26	24, 25	sylib 221	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))
27		id 23	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
28		elfvdm 6916	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
29	28	difexd 5302	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑘)) ∈ V)
30	29	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ (𝑀‘𝑘)) ∈ V)
31		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝑀‘𝑥) = (𝑀‘𝑘))
32	31	difeq2d 4089	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑋 ∖ (𝑀‘𝑥)) = (𝑋 ∖ (𝑀‘𝑘)))
33		eqid 2769	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))
34	32, 33	fvmptg 6988	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀‘𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (𝑋 ∖ (𝑀‘𝑘)))
35	27, 30, 34	syl2anr 608	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (𝑋 ∖ (𝑀‘𝑘)))
36	15	difeq1d 4088	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑘)) = (∪ 𝐽 ∖ (𝑀‘𝑘)))
37	35, 36	eqtrd 2804	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (∪ 𝐽 ∖ (𝑀‘𝑘)))
38	37	fveq2d 6886	. . . . . . . . 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 2807	. . . . . . . 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 2771	. . . . . . 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 247	. . . . . 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 243	. . . . 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 3183	. . . 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 591	. . 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 7077	. . . . . . . . 9 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑥 ∈ ℕ) → (𝑀‘𝑥) ∈ 𝐽)
46	14	difeq1d 4088	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑥)) = (∪ 𝐽 ∖ (𝑀‘𝑥)))
47	46	adantr 485	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀‘𝑥)) = (∪ 𝐽 ∖ (𝑀‘𝑥)))
48	11	opncld 23158	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
49	5, 48	sylan 591	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
50	47, 49	eqeltrd 2869	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
51	45, 50	sylan2 604	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀:ℕ⟶𝐽 ∧ 𝑥 ∈ ℕ)) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
52	51	anassrs 472	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
53	52	ralrimiva 3163	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
54	3, 53	sylan 591	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
55	33	fmpt 7106	. . . . 5 ⊢ (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽) ↔ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽))
56	54, 55	sylib 221	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽))
57		nne 2968	. . . . . . 7 ⊢ (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
58	57	ralbii 3117	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
59		ralnex 3097	. . . . . 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 280	. . . . 5 ⊢ (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅)
61	4	bcth 25456	. . . . . . 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 1137	. . . . . 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 2982	. . . . 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 245	. . . 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 602	. . 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 4083	. . . . 5 ⊢ (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅ → (∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = (∪ 𝐽 ∖ ∅))
67	28	difexd 5302	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
68	67	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
69	68	ralrimiva 3163	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
70	33	fnmpt 6676	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) Fn ℕ)
71		fniunfv 7246	. . . . . . . . . . . . 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 19	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))
73	35	iuneq2dv 4985	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘)))
74	32	cbviunv 5007	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘))
75	73, 74	eqtr4di 2822	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)))
76	72, 75	eqtr3d 2806	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)))
77		iundif2 5042	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) = (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥))
78	76, 77	eqtrdi 2820	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥)))
79		ffn 6706	. . . . . . . . . . . . 13 ⊢ (𝑀:ℕ⟶𝐽 → 𝑀 Fn ℕ)
80	79	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
81		fniinfv 6960	. . . . . . . . . . . 12 ⊢ (𝑀 Fn ℕ → ∩ 𝑥 ∈ ℕ (𝑀‘𝑥) = ∩ ran 𝑀)
82	80, 81	syl 18	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ 𝑥 ∈ ℕ (𝑀‘𝑥) = ∩ ran 𝑀)
83	82	difeq2d 4089	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥)) = (𝑋 ∖ ∩ ran 𝑀))
84	14	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑋 = ∪ 𝐽)
85	84	difeq1d 4088	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ ∩ ran 𝑀) = (∪ 𝐽 ∖ ∩ ran 𝑀))
86	78, 83, 85	3eqtrd 2808	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (∪ 𝐽 ∖ ∩ ran 𝑀))
87	86	fveq2d 6886	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀)))
88	87	difeq2d 4089	. . . . . . 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 485	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝐽 ∈ Top)
90		1nn 12243	. . . . . . . . 9 ⊢ 1 ∈ ℕ
91		biidd 265	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽) ↔ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽)))
92		fnfvelrn 7076	. . . . . . . . . . . . . 14 ⊢ ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
93	80, 92	sylan 591	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
94		intss1 4932	. . . . . . . . . . . . 13 ⊢ ((𝑀‘𝑘) ∈ ran 𝑀 → ∩ ran 𝑀 ⊆ (𝑀‘𝑘))
95	93, 94	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ∩ ran 𝑀 ⊆ (𝑀‘𝑘))
96	95, 10	sstrd 3955	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ∩ ran 𝑀 ⊆ ∪ 𝐽)
97	96	expcom 418	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽))
98	91, 97	vtoclga 3550	. . . . . . . . 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 23175	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ∩ ran 𝑀 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
101	89, 99, 100	syl2anc 595	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
102	88, 101	eqtr4d 2807	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = ((cls‘𝐽)‘∩ ran 𝑀))
103		dif0 4341	. . . . . . 7 ⊢ (∪ 𝐽 ∖ ∅) = ∪ 𝐽
104	103, 84	eqtr4id 2823	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ∅) = 𝑋)
105	102, 104	eqeq12d 2785	. . . . 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 247	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅ → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋))
107	3, 106	sylan 591	. . 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 61	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋 → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋))
109	108	3impia 1133	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 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  ∪ cuni 4876  ∩ cint 4916  ∪ ciun 4960  ∩ ciin 4961   ↦ cmpt 5196  dom cdm 5662  ran crn 5663   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  1c1 11100  ℕcn 12232  ∞Metcxmet 21475  Metcmet 21476  MetOpencmopn 21480  Topctop 23018  Clsdccld 23141  intcnt 23142  clsccl 23143  CMetccmet 25381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9609  ax-dc 10429  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9401  df-inf 9402  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-n0 12504  df-z 12591  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ico 13377  df-rest 17474  df-topgen 17495  df-psmet 21482  df-xmet 21483  df-met 21484  df-bl 21485  df-mopn 21486  df-fbas 21487  df-fg 21488  df-top 23019  df-topon 23036  df-bases 23071  df-cld 23144  df-ntr 23145  df-cls 23146  df-nei 23223  df-lm 23354  df-fil 23971  df-fm 24063  df-flim 24064  df-flf 24065  df-cfil 25382  df-cau 25383  df-cmet 25384

This theorem is referenced by: (None)