Theorem bcth3 25652

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 25607	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2		metxmet 24653	. . . . 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 24759	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
6	5	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7		ffvelcdm 7081	. . . . . . . . . 10 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ 𝐽)
8		elssuni 4899	. . . . . . . . . 10 ⊢ ((𝑀‘𝑘) ∈ 𝐽 → (𝑀‘𝑘) ⊆ ∪ 𝐽)
9	7, 8	syl 18	. . . . . . . . 9 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
10	9	adantll 727	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
11		eqid 2761	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
12	11	clsval2 23368	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑘) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
13	6, 10, 12	syl2anc 596	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
14	4	mopnuni 24760	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
15	14	ad2antrr 739	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = ∪ 𝐽)
16	13, 15	eqeq12d 2777	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋 ↔ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽))
17		difeq2 4068	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = (∪ 𝐽 ∖ ∪ 𝐽))
18		difid 4325	. . . . . . . 8 ⊢ (∪ 𝐽 ∖ ∪ 𝐽) = ∅
19	17, 18	eqtrdi 2812	. . . . . . 7 ⊢ ((∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ∅)
20		difss 4083	. . . . . . . . . . . 12 ⊢ (∪ 𝐽 ∖ (𝑀‘𝑘)) ⊆ ∪ 𝐽
21	11	ntropn 23367	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ (𝑀‘𝑘)) ⊆ ∪ 𝐽) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
22	6, 20, 21	sylancl 598	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
23		elssuni 4899	. . . . . . . . . . 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 4215	. . . . . . . . . 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 6919	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
29	28	difexd 5293	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑘)) ∈ V)
30	29	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ (𝑀‘𝑘)) ∈ V)
31		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝑀‘𝑥) = (𝑀‘𝑘))
32	31	difeq2d 4074	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑋 ∖ (𝑀‘𝑥)) = (𝑋 ∖ (𝑀‘𝑘)))
33		eqid 2761	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))
34	32, 33	fvmptg 6991	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀‘𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (𝑋 ∖ (𝑀‘𝑘)))
35	27, 30, 34	syl2anr 609	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (𝑋 ∖ (𝑀‘𝑘)))
36	15	difeq1d 4073	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑘)) = (∪ 𝐽 ∖ (𝑀‘𝑘)))
37	35, 36	eqtrd 2796	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (∪ 𝐽 ∖ (𝑀‘𝑘)))
38	37	fveq2d 6889	. . . . . . . . 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 2799	. . . . . . . 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 2763	. . . . . . 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 3175	. . . 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 592	. . 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 7081	. . . . . . . . 9 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑥 ∈ ℕ) → (𝑀‘𝑥) ∈ 𝐽)
46	14	difeq1d 4073	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑥)) = (∪ 𝐽 ∖ (𝑀‘𝑥)))
47	46	adantr 486	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀‘𝑥)) = (∪ 𝐽 ∖ (𝑀‘𝑥)))
48	11	opncld 23351	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
49	5, 48	sylan 592	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
50	47, 49	eqeltrd 2861	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
51	45, 50	sylan2 605	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀:ℕ⟶𝐽 ∧ 𝑥 ∈ ℕ)) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
52	51	anassrs 473	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
53	52	ralrimiva 3155	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
54	3, 53	sylan 592	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
55	33	fmpt 7110	. . . . 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 2960	. . . . . . 7 ⊢ (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
58	57	ralbii 3109	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
59		ralnex 3089	. . . . . 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 25650	. . . . . . 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 1139	. . . . . 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 2974	. . . . 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 603	. . 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 4068	. . . . 5 ⊢ (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅ → (∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = (∪ 𝐽 ∖ ∅))
67	28	difexd 5293	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
68	67	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
69	68	ralrimiva 3155	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
70	33	fnmpt 6679	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) Fn ℕ)
71		fniunfv 7251	. . . . . . . . . . . . 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 4976	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘)))
74	32	cbviunv 4997	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘))
75	73, 74	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)))
76	72, 75	eqtr3d 2798	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)))
77		iundif2 5032	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) = (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥))
78	76, 77	eqtrdi 2812	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥)))
79		ffn 6709	. . . . . . . . . . . . 13 ⊢ (𝑀:ℕ⟶𝐽 → 𝑀 Fn ℕ)
80	79	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
81		fniinfv 6963	. . . . . . . . . . . 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 4074	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥)) = (𝑋 ∖ ∩ ran 𝑀))
84	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑋 = ∪ 𝐽)
85	84	difeq1d 4073	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ ∩ ran 𝑀) = (∪ 𝐽 ∖ ∩ ran 𝑀))
86	78, 83, 85	3eqtrd 2800	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (∪ 𝐽 ∖ ∩ ran 𝑀))
87	86	fveq2d 6889	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀)))
88	87	difeq2d 4074	. . . . . . 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 486	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝐽 ∈ Top)
90		1nn 12346	. . . . . . . . 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 7080	. . . . . . . . . . . . . 14 ⊢ ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
93	80, 92	sylan 592	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
94		intss1 4923	. . . . . . . . . . . . 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 3941	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ∩ ran 𝑀 ⊆ ∪ 𝐽)
97	96	expcom 419	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽))
98	91, 97	vtoclga 3537	. . . . . . . . 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 23368	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ∩ ran 𝑀 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
101	89, 99, 100	syl2anc 596	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
102	88, 101	eqtr4d 2799	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = ((cls‘𝐽)‘∩ ran 𝑀))
103		dif0 4327	. . . . . . 7 ⊢ (∪ 𝐽 ∖ ∅) = ∪ 𝐽
104	103, 84	eqtr4id 2815	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ∅) = 𝑋)
105	102, 104	eqeq12d 2777	. . . . 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 592	. . 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 1135	1 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋) → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867  ∩ cint 4907  ∪ ciun 4951  ∩ ciin 4952   ↦ cmpt 5186  dom cdm 5651  ran crn 5652   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  1c1 11201  ℕcn 12335  ∞Metcxmet 21663  Metcmet 21664  MetOpencmopn 21668  Topctop 23211  Clsdccld 23334  intcnt 23335  clsccl 23336  CMetccmet 25575

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642  ax-dc 10524  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-ico 13482  df-rest 17593  df-topgen 17614  df-psmet 21670  df-xmet 21671  df-met 21672  df-bl 21673  df-mopn 21674  df-fbas 21675  df-fg 21676  df-top 23212  df-topon 23229  df-bases 23264  df-cld 23337  df-ntr 23338  df-cls 23339  df-nei 23416  df-lm 23547  df-fil 24165  df-fm 24257  df-flim 24258  df-flf 24259  df-cfil 25576  df-cau 25577  df-cmet 25578

This theorem is used by: (None)