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Theorem met1stc 24496

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 24415	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
3	1	mopnuni 24416	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
4	3	eleq2d 2823	. . . . 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 12945	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
9	8	adantl 481	. . . . . . . . . . 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 24475	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ^*) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
13	6, 7, 11, 12	syl3anc 1374	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
14	13	fmpttd 7061	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ⟶𝐽)
15	14	frnd 6670	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ⊆ 𝐽)
16		nnex 12171	. . . . . . . . 9 ⊢ ℕ ∈ V
17	16	mptex 7171	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V
18	17	rnex 7854	. . . . . . 7 ⊢ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V
19	18	elpw 4546	. . . . . 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 9558	. . . . . . . . 9 ⊢ ω ∈ On
22		nnenom 13933	. . . . . . . . . 10 ⊢ ℕ ≈ ω
23	22	ensymi 8944	. . . . . . . . 9 ⊢ ω ≈ ℕ
24		isnumi 9861	. . . . . . . . 9 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
25	21, 23, 24	mp2an 693	. . . . . . . 8 ⊢ ℕ ∈ dom card
26		ovex 7393	. . . . . . . . . 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 6635	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) Fn ℕ
29		dffn4 6752	. . . . . . . . 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 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 8953	. . . . . . 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 693	. . . . . 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 24468	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)
40	36, 37, 38, 39	syl3anc 1374	. . . . . . . . 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 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 24388	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑦) ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)))
47	41, 42, 45, 46	syl3anc 1374	. . . . . . . . . . 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 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 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 729	. . . . . . . . . . . . . 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 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 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 24398	. . . . . . . . . . . . 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 1384	. . . . . . . . . . . 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 3933	. . . . . . . . . . 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 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 217	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ⁺ → ∃𝑦 ∈ ℕ (1 / 𝑦) < 𝑟)
64	63	ad2antrl 729	. . . . . . . . . 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 3154	. . . . . . . . 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 3145	. . . . . . . 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 7395	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ 𝑦 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑦)) ∈ V)
68		vex 3434	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
69		oveq2 7368	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑦 → (1 / 𝑛) = (1 / 𝑦))
70	69	oveq2d 7376	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑦 → (𝑥(ball‘𝐷)(1 / 𝑛)) = (𝑥(ball‘𝐷)(1 / 𝑦)))
71	70	cbvmptv 5190	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) = (𝑦 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑦)))
72	71	elrnmpt 5907	. . . . . . . . . 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 2826	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦))))
75		sseq1 3948	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑤 ⊆ 𝑧 ↔ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧))
76	74, 75	anbi12d 633	. . . . . . . . . 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 5348	. . . . . . . 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 3130	. . . . 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 5089	. . . . . . 7 ⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → (𝑦 ≼ ω ↔ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω))
83		rexeq 3292	. . . . . . . . 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 3161	. . . . . . 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 633	. . . . . 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 3565	. . . . 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 592	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
90	89	ralrimiva 3130	. 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 23417	. 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 584	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1^stω)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3430 ⊆ wss 3890 𝒫 cpw 4542 ∪ cuni 4851 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5624 ran crn 5625 Oncon0 6317 Fn wfn 6487 –onto→wfo 6490 ‘cfv 6492 (class class class)co 7360 ωcom 7810 ≈ cen 8883 ≼ cdom 8884 cardccrd 9850 ℝcr 11028 0cc0 11029 1c1 11030 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 / cdiv 11798 ℕcn 12165 ℝ⁺crp 12933 ∞Metcxmet 21329 ballcbl 21331 MetOpencmopn 21334 Topctop 22868 1^stωc1stc 23412

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 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 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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9348 df-inf 9349 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 21336 df-xmet 21337 df-bl 21339 df-mopn 21340 df-top 22869 df-topon 22886 df-bases 22921 df-1stc 23414

This theorem is referenced by: metelcls 25282 metcnp4 25287 metcn4 25288

W3C validator