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 24504

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 24423	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
3	1	mopnuni 24424	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
4	3	eleq2d 2825	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
5	4	biimpar 478	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋)
6		simpll 772	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
7		simplr 774	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝑋)
8		nnrp 12945	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
9	8	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
10	9	rpreccld 12987	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
11	10	rpxrd 12978	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ^*)
12	1	blopn 24483	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ^*) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
13	6, 7, 11, 12	syl3anc 1379	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
14	13	fmpttd 7056	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ⟶𝐽)
15	14	frnd 6663	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ⊆ 𝐽)
16		nnex 12171	. . . . . . . . 9 ⊢ ℕ ∈ V
17	16	mptex 7167	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V
18	17	rnex 7850	. . . . . . 7 ⊢ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V
19	18	elpw 4533	. . . . . 6 ⊢ (ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽 ↔ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ⊆ 𝐽)
20	15, 19	sylibr 235	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽)
21		omelon 9558	. . . . . . . . 9 ⊢ ω ∈ On
22		nnenom 13933	. . . . . . . . . 10 ⊢ ℕ ≈ ω
23	22	ensymi 8941	. . . . . . . . 9 ⊢ ω ≈ ℕ
24		isnumi 9861	. . . . . . . . 9 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
25	21, 23, 24	mp2an 698	. . . . . . . 8 ⊢ ℕ ∈ dom card
26		ovex 7389	. . . . . . . . . 10 ⊢ (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ V
27		eqid 2739	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))
28	26, 27	fnmpti 6628	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) Fn ℕ
29		dffn4 6745	. . . . . . . . 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 231	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ–onto→ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))
31		fodomnum 9970	. . . . . . . 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 8950	. . . . . . 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 698	. . . . . 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 772	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝐷 ∈ (∞Met‘𝑋))
37		simprl 776	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝑧 ∈ 𝐽)
38		simprr 778	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ 𝑧)
39	1	mopni2 24476	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)
40	36, 37, 38, 39	syl3anc 1379	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)
41		simp-4l 788	. . . . . . . . . . . 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 789	. . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 12975	. . . . . . . . . . . . 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 12987	. . . . . . . . . . . 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 24396	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑦) ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)))
47	41, 42, 45, 46	syl3anc 1379	. . . . . . . . . . 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 12978	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . . 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 12978	. . . . . . . . . . . . 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 12210	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ → (1 / 𝑦) ∈ ℝ)
52	51	ad2antrl 734	. . . . . . . . . . . . . 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 12977	. . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . 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 11285	. . . . . . . . . . . . 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 24406	. . . . . . . . . . . . 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 1389	. . . . . . . . . . . 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 783	. . . . . . . . . . . 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 3925	. . . . . . . . . . 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 516	. . . . . . . . . 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 12935	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
62		nnrecl 12426	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑦 ∈ ℕ (1 / 𝑦) < 𝑟)
63	61, 62	sylbi 218	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → ∃𝑦 ∈ ℕ (1 / 𝑦) < 𝑟)
64	63	ad2antrl 734	. . . . . . . . . 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 3155	. . . . . . . . 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 3146	. . . . . . . 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 7391	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ 𝑦 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑦)) ∈ V)
68		vex 3435	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
69		oveq2 7364	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑦 → (1 / 𝑛) = (1 / 𝑦))
70	69	oveq2d 7372	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑦 → (𝑥(ball‘𝐷)(1 / 𝑛)) = (𝑥(ball‘𝐷)(1 / 𝑦)))
71	70	cbvmptv 5176	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) = (𝑦 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑦)))
72	71	elrnmpt 5900	. . . . . . . . . 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 2828	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦))))
75		sseq1 3940	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑤 ⊆ 𝑧 ↔ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧))
76	74, 75	anbi12d 638	. . . . . . . . . 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 482	. . . . . . . . 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 5340	. . . . . . . 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 258	. . . . . . 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 457	. . . . . 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 3131	. . . . 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 5075	. . . . . . 7 ⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → (𝑦 ≼ ω ↔ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω))
83		rexeq 3293	. . . . . . . . 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 341	. . . . . . . 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 3162	. . . . . . 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 638	. . . . . 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 3560	. . . . 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 842	. . . 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 597	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
90	89	ralrimiva 3131	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
91		eqid 2739	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
92	91	is1stc2 23425	. 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 589	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1^stω)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ⊆ wss 3883 𝒫 cpw 4529 ∪ cuni 4838 class class class wbr 5072 ↦ cmpt 5153 dom cdm 5618 ran crn 5619 Oncon0 6310 Fn wfn 6480 –onto→wfo 6483 ‘cfv 6485 (class class class)co 7356 ωcom 7806 ≈ cen 8880 ≼ cdom 8881 cardccrd 9850 ℝcr 11028 0cc0 11029 1c1 11030 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 / cdiv 11798 ℕcn 12165 ℝ⁺crp 12933 ∞Metcxmet 21332 ballcbl 21334 MetOpencmopn 21337 Topctop 22876 1^stωc1stc 23420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-n0 12429 df-z 12516 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-topgen 17397 df-psmet 21339 df-xmet 21340 df-bl 21342 df-mopn 21343 df-top 22877 df-topon 22894 df-bases 22929 df-1stc 23422

This theorem is referenced by: metelcls 25290 metcnp4 25295 metcn4 25296

W3C validator