met1stc - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > met1stc		Structured version Visualization version GIF version

Theorem met1stc 24534

Description: The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

**Hypothesis**
Ref	Expression
methaus.1	⊢ 𝐽 = (MetOpen‘𝐷)

**Assertion**
Ref	Expression
met1stc	⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1^stω)

Proof of Theorem met1stc Dummy variables 𝑛 𝑟 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		methaus.1	. . 3 ⊢ 𝐽 = (MetOpen‘𝐷)
2	1	mopntop 24450	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
3	1	mopnuni 24451	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
4	3	eleq2d 2827	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
5	4	biimpar 477	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋)
6		simpll 767	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
7		simplr 769	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝑋)
8		nnrp 13046	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
9	8	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
10	9	rpreccld 13087	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
11	10	rpxrd 13078	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ^*)
12	1	blopn 24513	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ^*) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
13	6, 7, 11, 12	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
14	13	fmpttd 7135	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ⟶𝐽)
15	14	frnd 6744	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ⊆ 𝐽)
16		nnex 12272	. . . . . . . . 9 ⊢ ℕ ∈ V
17	16	mptex 7243	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V
18	17	rnex 7932	. . . . . . 7 ⊢ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V
19	18	elpw 4604	. . . . . 6 ⊢ (ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽 ↔ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ⊆ 𝐽)
20	15, 19	sylibr 234	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽)
21		omelon 9686	. . . . . . . . 9 ⊢ ω ∈ On
22		nnenom 14021	. . . . . . . . . 10 ⊢ ℕ ≈ ω
23	22	ensymi 9044	. . . . . . . . 9 ⊢ ω ≈ ℕ
24		isnumi 9986	. . . . . . . . 9 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
25	21, 23, 24	mp2an 692	. . . . . . . 8 ⊢ ℕ ∈ dom card
26		ovex 7464	. . . . . . . . . 10 ⊢ (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ V
27		eqid 2737	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))
28	26, 27	fnmpti 6711	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) Fn ℕ
29		dffn4 6826	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) Fn ℕ ↔ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ–onto→ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))))
30	28, 29	mpbi 230	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ–onto→ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))
31		fodomnum 10097	. . . . . . . 8 ⊢ (ℕ ∈ dom card → ((𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ–onto→ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ℕ))
32	25, 30, 31	mp2 9	. . . . . . 7 ⊢ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ℕ
33		domentr 9053	. . . . . . 7 ⊢ ((ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ℕ ∧ ℕ ≈ ω) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω)
34	32, 22, 33	mp2an 692	. . . . . 6 ⊢ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω
35	34	a1i 11	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω)
36		simpll 767	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝐷 ∈ (∞Met‘𝑋))
37		simprl 771	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝑧 ∈ 𝐽)
38		simprr 773	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ 𝑧)
39	1	mopni2 24506	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)
40	36, 37, 38, 39	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)
41		simp-4l 783	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝐷 ∈ (∞Met‘𝑋))
42		simp-4r 784	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑥 ∈ 𝑋)
43		simprl 771	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑦 ∈ ℕ)
44	43	nnrpd 13075	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑦 ∈ ℝ⁺)
45	44	rpreccld 13087	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) ∈ ℝ⁺)
46		blcntr 24423	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑦) ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)))
47	41, 42, 45, 46	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)))
48	45	rpxrd 13078	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) ∈ ℝ^*)
49		simplrl 777	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑟 ∈ ℝ⁺)
50	49	rpxrd 13078	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑟 ∈ ℝ^*)
51		nnrecre 12308	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ → (1 / 𝑦) ∈ ℝ)
52	51	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) ∈ ℝ)
53	49	rpred 13077	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑟 ∈ ℝ)
54		simprr 773	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) < 𝑟)
55	52, 53, 54	ltled 11409	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) ≤ 𝑟)
56		ssbl 24433	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ ((1 / 𝑦) ∈ ℝ^* ∧ 𝑟 ∈ ℝ^*) ∧ (1 / 𝑦) ≤ 𝑟) → (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ (𝑥(ball‘𝐷)𝑟))
57	41, 42, 48, 50, 55, 56	syl221anc 1383	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ (𝑥(ball‘𝐷)𝑟))
58		simplrr 778	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)
59	57, 58	sstrd 3994	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧)
60	47, 59	jca 511	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧))
61		elrp 13036	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
62		nnrecl 12524	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑦 ∈ ℕ (1 / 𝑦) < 𝑟)
63	61, 62	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → ∃𝑦 ∈ ℕ (1 / 𝑦) < 𝑟)
64	63	ad2antrl 728	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) → ∃𝑦 ∈ ℕ (1 / 𝑦) < 𝑟)
65	60, 64	reximddv 3171	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) → ∃𝑦 ∈ ℕ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧))
66	40, 65	rexlimddv 3161	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → ∃𝑦 ∈ ℕ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧))
67		ovexd 7466	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ 𝑦 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑦)) ∈ V)
68		vex 3484	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
69		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑦 → (1 / 𝑛) = (1 / 𝑦))
70	69	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑦 → (𝑥(ball‘𝐷)(1 / 𝑛)) = (𝑥(ball‘𝐷)(1 / 𝑦)))
71	70	cbvmptv 5255	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) = (𝑦 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑦)))
72	71	elrnmpt 5969	. . . . . . . . . 10 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ↔ ∃𝑦 ∈ ℕ 𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦))))
73	68, 72	mp1i 13	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → (𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ↔ ∃𝑦 ∈ ℕ 𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦))))
74		eleq2 2830	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦))))
75		sseq1 4009	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑤 ⊆ 𝑧 ↔ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧))
76	74, 75	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧)))
77	76	adantl 481	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ 𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦))) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧)))
78	67, 73, 77	rexxfr2d 5411	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → (∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑦 ∈ ℕ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧)))
79	66, 78	mpbird 257	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
80	79	expr 456	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝐽) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
81	80	ralrimiva 3146	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
82		breq1 5146	. . . . . . 7 ⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → (𝑦 ≼ ω ↔ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω))
83		rexeq 3322	. . . . . . . . 9 ⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → (∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
84	83	imbi2d 340	. . . . . . . 8 ⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → ((𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
85	84	ralbidv 3178	. . . . . . 7 ⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
86	82, 85	anbi12d 632	. . . . . 6 ⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ↔ (ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))
87	86	rspcev 3622	. . . . 5 ⊢ ((ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽 ∧ (ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) → ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
88	20, 35, 81, 87	syl12anc 837	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
89	5, 88	syldan 591	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
90	89	ralrimiva 3146	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
91		eqid 2737	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
92	91	is1stc2 23450	. 2 ⊢ (𝐽 ∈ 1^stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))
93	2, 90, 92	sylanbrc 583	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1^stω)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 Oncon0 6384 Fn wfn 6556 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 cardccrd 9975 ℝcr 11154 0cc0 11155 1c1 11156 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 / cdiv 11920 ℕcn 12266 ℝ⁺crp 13034 ∞Metcxmet 21349 ballcbl 21351 MetOpencmopn 21354 Topctop 22899 1^stωc1stc 23445

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-1stc 23447

This theorem is referenced by: metelcls 25339 metcnp4 25344 metcn4 25345

W3C validator