Theorem bcth3 25299

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 25254	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2		metxmet 24290	. . . . 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 24396	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
6	5	ad2antrr 727	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7		ffvelcdm 7035	. . . . . . . . . 10 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ 𝐽)
8		elssuni 4896	. . . . . . . . . 10 ⊢ ((𝑀‘𝑘) ∈ 𝐽 → (𝑀‘𝑘) ⊆ ∪ 𝐽)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
10	9	adantll 715	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
11		eqid 2737	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
12	11	clsval2 23006	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑘) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
13	6, 10, 12	syl2anc 585	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
14	4	mopnuni 24397	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
15	14	ad2antrr 727	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = ∪ 𝐽)
16	13, 15	eqeq12d 2753	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (((cls‘𝐽)‘(𝑀‘𝑘)) = 𝑋 ↔ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽))
17		difeq2 4074	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = (∪ 𝐽 ∖ ∪ 𝐽))
18		difid 4330	. . . . . . . 8 ⊢ (∪ 𝐽 ∖ ∪ 𝐽) = ∅
19	17, 18	eqtrdi 2788	. . . . . . 7 ⊢ ((∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))) = ∪ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ∅)
20		difss 4090	. . . . . . . . . . . 12 ⊢ (∪ 𝐽 ∖ (𝑀‘𝑘)) ⊆ ∪ 𝐽
21	11	ntropn 23005	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ (𝑀‘𝑘)) ⊆ ∪ 𝐽) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
22	6, 20, 21	sylancl 587	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
23		elssuni 4896	. . . . . . . . . . 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 4223	. . . . . . . . . 10 ⊢ (((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))
26	24, 25	sylib 218	. . . . . . . . 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 6876	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
29	28	difexd 5278	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑘)) ∈ V)
30	29	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ (𝑀‘𝑘)) ∈ V)
31		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝑀‘𝑥) = (𝑀‘𝑘))
32	31	difeq2d 4080	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑋 ∖ (𝑀‘𝑥)) = (𝑋 ∖ (𝑀‘𝑘)))
33		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))
34	32, 33	fvmptg 6947	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀‘𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (𝑋 ∖ (𝑀‘𝑘)))
35	27, 30, 34	syl2anr 598	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (𝑋 ∖ (𝑀‘𝑘)))
36	15	difeq1d 4079	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑘)) = (∪ 𝐽 ∖ (𝑀‘𝑘)))
37	35, 36	eqtrd 2772	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (∪ 𝐽 ∖ (𝑀‘𝑘)))
38	37	fveq2d 6846	. . . . . . . . 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 2775	. . . . . . . 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 2739	. . . . . . 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 244	. . . . . 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 240	. . . . 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 3150	. . . 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 581	. . 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 7035	. . . . . . . . 9 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑥 ∈ ℕ) → (𝑀‘𝑥) ∈ 𝐽)
46	14	difeq1d 4079	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑥)) = (∪ 𝐽 ∖ (𝑀‘𝑥)))
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀‘𝑥)) = (∪ 𝐽 ∖ (𝑀‘𝑥)))
48	11	opncld 22989	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
49	5, 48	sylan 581	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
50	47, 49	eqeltrd 2837	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀‘𝑥) ∈ 𝐽) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
51	45, 50	sylan2 594	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑀:ℕ⟶𝐽 ∧ 𝑥 ∈ ℕ)) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
52	51	anassrs 467	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
53	52	ralrimiva 3130	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
54	3, 53	sylan 581	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
55	33	fmpt 7064	. . . . 5 ⊢ (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽) ↔ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽))
56	54, 55	sylib 218	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽))
57		nne 2937	. . . . . . 7 ⊢ (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
58	57	ralbii 3084	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
59		ralnex 3064	. . . . . 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 277	. . . . 5 ⊢ (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅)
61	4	bcth 25297	. . . . . . 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 1122	. . . . . 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 2951	. . . . 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 242	. . . 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 592	. . 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 4074	. . . . 5 ⊢ (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅ → (∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = (∪ 𝐽 ∖ ∅))
67	28	difexd 5278	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
68	67	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
69	68	ralrimiva 3130	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
70	33	fnmpt 6640	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) Fn ℕ)
71		fniunfv 7203	. . . . . . . . . . . . 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 4973	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘)))
74	32	cbviunv 4996	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘))
75	73, 74	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)))
76	72, 75	eqtr3d 2774	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)))
77		iundif2 5031	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) = (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥))
78	76, 77	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥)))
79		ffn 6670	. . . . . . . . . . . . 13 ⊢ (𝑀:ℕ⟶𝐽 → 𝑀 Fn ℕ)
80	79	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
81		fniinfv 6920	. . . . . . . . . . . 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 4080	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥)) = (𝑋 ∖ ∩ ran 𝑀))
84	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑋 = ∪ 𝐽)
85	84	difeq1d 4079	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑋 ∖ ∩ ran 𝑀) = (∪ 𝐽 ∖ ∩ ran 𝑀))
86	78, 83, 85	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (∪ 𝐽 ∖ ∩ ran 𝑀))
87	86	fveq2d 6846	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀)))
88	87	difeq2d 4080	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝐽 ∈ Top)
90		1nn 12168	. . . . . . . . 9 ⊢ 1 ∈ ℕ
91		biidd 262	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽) ↔ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽)))
92		fnfvelrn 7034	. . . . . . . . . . . . . 14 ⊢ ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
93	80, 92	sylan 581	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
94		intss1 4920	. . . . . . . . . . . . 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 3946	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ∩ ran 𝑀 ⊆ ∪ 𝐽)
97	96	expcom 413	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽))
98	91, 97	vtoclga 3534	. . . . . . . . 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 23006	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ∩ ran 𝑀 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
101	89, 99, 100	syl2anc 585	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
102	88, 101	eqtr4d 2775	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))))) = ((cls‘𝐽)‘∩ ran 𝑀))
103		dif0 4332	. . . . . . 7 ⊢ (∪ 𝐽 ∖ ∅) = ∪ 𝐽
104	103, 84	eqtr4id 2791	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ∅) = 𝑋)
105	102, 104	eqeq12d 2753	. . . . 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 244	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅ → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋))
107	3, 106	sylan 581	. . 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 1118	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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  ∪ cuni 4865  ∩ cint 4904  ∪ ciun 4948  ∩ ciin 4949   ↦ cmpt 5181  dom cdm 5632  ran crn 5633   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  1c1 11039  ℕcn 12157  ∞Metcxmet 21306  Metcmet 21307  MetOpencmopn 21311  Topctop 22849  Clsdccld 22972  intcnt 22973  clsccl 22974  CMetccmet 25222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-dc 10368  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9357  df-inf 9358  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ico 13279  df-rest 17354  df-topgen 17375  df-psmet 21313  df-xmet 21314  df-met 21315  df-bl 21316  df-mopn 21317  df-fbas 21318  df-fg 21319  df-top 22850  df-topon 22867  df-bases 22902  df-cld 22975  df-ntr 22976  df-cls 22977  df-nei 23054  df-lm 23185  df-fil 23802  df-fm 23894  df-flim 23895  df-flf 23896  df-cfil 25223  df-cau 25224  df-cmet 25225

This theorem is referenced by: (None)