Theorem bcth3 24495

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 24450	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
2		metxmet 23487	. . . . 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 23593	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
6	5	ad2antrr 723	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝐽 ∈ Top)
7		ffvelrn 6959	. . . . . . . . . 10 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ 𝐽)
8		elssuni 4871	. . . . . . . . . 10 ⊢ ((𝑀‘𝑘) ∈ 𝐽 → (𝑀‘𝑘) ⊆ ∪ 𝐽)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ ((𝑀:ℕ⟶𝐽 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
10	9	adantll 711	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
11		eqid 2738	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
12	11	clsval2 22201	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑘) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
13	6, 10, 12	syl2anc 584	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((cls‘𝐽)‘(𝑀‘𝑘)) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘)))))
14	4	mopnuni 23594	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
15	14	ad2antrr 723	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → 𝑋 = ∪ 𝐽)
16	13, 15	eqeq12d 2754	. . . . . 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 2794	. . . . . . 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 22200	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ (𝑀‘𝑘)) ⊆ ∪ 𝐽) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
22	6, 20, 21	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ∈ 𝐽)
23		elssuni 4871	. . . . . . . . . . 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 4192	. . . . . . . . . 10 ⊢ (((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))) ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))) = ((int‘𝐽)‘(∪ 𝐽 ∖ (𝑀‘𝑘))))
26	24, 25	sylib 217	. . . . . . . . 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 6806	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
29	28	difexd 5253	. . . . . . . . . . . . 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 6774	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝑀‘𝑥) = (𝑀‘𝑘))
32	31	difeq2d 4057	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (𝑋 ∖ (𝑀‘𝑥)) = (𝑋 ∖ (𝑀‘𝑘)))
33		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))
34	32, 33	fvmptg 6873	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ (𝑋 ∖ (𝑀‘𝑘)) ∈ V) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (𝑋 ∖ (𝑀‘𝑘)))
35	27, 30, 34	syl2anr 597	. . . . . . . . . . 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 2778	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = (∪ 𝐽 ∖ (𝑀‘𝑘)))
38	37	fveq2d 6778	. . . . . . . . 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 2781	. . . . . . . 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 2740	. . . . . . 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	syl5ib 243	. . . . . 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 239	. . . . 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 3108	. . . 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 580	. . 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		ffvelrn 6959	. . . . . . . . 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 22184	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑀‘𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
49	5, 48	sylan 580	. . . . . . . . . 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 593	. . . . . . . 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 3103	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
54	3, 53	sylan 580	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽))
55	33	fmpt 6984	. . . . 5 ⊢ (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ (Clsd‘𝐽) ↔ (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽))
56	54, 55	sylib 217	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))):ℕ⟶(Clsd‘𝐽))
57		nne 2947	. . . . . . 7 ⊢ (¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
58	57	ralbii 3092	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ¬ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅ ↔ ∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅)
59		ralnex 3167	. . . . . 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 276	. . . . 5 ⊢ (∀𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) = ∅ ↔ ¬ ∃𝑘 ∈ ℕ ((int‘𝐽)‘((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘)) ≠ ∅)
61	4	bcth 24493	. . . . . . 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 1120	. . . . . 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 2961	. . . . 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	syl5bi 241	. . . 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 591	. . 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 5253	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
68	67	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑥 ∈ ℕ) → (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
69	68	ralrimiva 3103	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V)
70	33	fnmpt 6573	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) ∈ V → (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) Fn ℕ)
71		fniunfv 7120	. . . . . . . . . . . . 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 4948	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘)))
74	32	cbviunv 4970	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)) = ∪ 𝑘 ∈ ℕ (𝑋 ∖ (𝑀‘𝑘))
75	73, 74	eqtr4di 2796	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ 𝑘 ∈ ℕ ((𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))‘𝑘) = ∪ 𝑥 ∈ ℕ (𝑋 ∖ (𝑀‘𝑥)))
76	72, 75	eqtr3d 2780	. . . . . . . . . . 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 2794	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (𝑋 ∖ ∩ 𝑥 ∈ ℕ (𝑀‘𝑥)))
79		ffn 6600	. . . . . . . . . . . . 13 ⊢ (𝑀:ℕ⟶𝐽 → 𝑀 Fn ℕ)
80	79	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → 𝑀 Fn ℕ)
81		fniinfv 6846	. . . . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥))) = (∪ 𝐽 ∖ ∩ ran 𝑀))
87	86	fveq2d 6778	. . . . . . . 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 11984	. . . . . . . . 9 ⊢ 1 ∈ ℕ
91		biidd 261	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽) ↔ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ∩ ran 𝑀 ⊆ ∪ 𝐽)))
92		fnfvelrn 6958	. . . . . . . . . . . . . 14 ⊢ ((𝑀 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
93	80, 92	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ ran 𝑀)
94		intss1 4894	. . . . . . . . . . . . 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 3931	. . . . . . . . . . 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 3513	. . . . . . . . 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 22201	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ∩ ran 𝑀 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
101	89, 99, 100	syl2anc 584	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → ((cls‘𝐽)‘∩ ran 𝑀) = (∪ 𝐽 ∖ ((int‘𝐽)‘(∪ 𝐽 ∖ ∩ ran 𝑀))))
102	88, 101	eqtr4d 2781	. . . . . 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 2797	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (∪ 𝐽 ∖ ∅) = 𝑋)
105	102, 104	eqeq12d 2754	. . . . 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	syl5ib 243	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀:ℕ⟶𝐽) → (((int‘𝐽)‘∪ ran (𝑥 ∈ ℕ ↦ (𝑋 ∖ (𝑀‘𝑥)))) = ∅ → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋))
107	3, 106	sylan 580	. . 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 1116	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 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839  ∩ cint 4879  ∪ ciun 4924  ∩ ciin 4925   ↦ cmpt 5157  dom cdm 5589  ran crn 5590   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  1c1 10872  ℕcn 11973  ∞Metcxmet 20582  Metcmet 20583  MetOpencmopn 20587  Topctop 22042  Clsdccld 22167  intcnt 22168  clsccl 22169  CMetccmet 24418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-dc 10202  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ico 13085  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lm 22380  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-cfil 24419  df-cau 24420  df-cmet 24421

This theorem is referenced by: (None)