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Theorem met2ndci 22327

Description: A separable metric space (a metric space with a countable dense subset) is second-countable. (Contributed by Mario Carneiro, 13-Apr-2015.)

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

**Assertion**
Ref	Expression
met2ndci	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ 2^nd𝜔)

Proof of Theorem met2ndci Dummy variables 𝑛 𝑟 𝑡 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		methaus.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷)
2	1	mopntop 22245	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
3	2	adantr 481	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ Top)
4		simpll 790	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
5		simplr1 1103	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝐴 ⊆ 𝑋)
6		simprr 796	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
7	5, 6	sseldd 3604	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝑋)
8		simprl 794	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ)
9	8	nnrpd 11870	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ⁺)
10	9	rpreccld 11882	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (1 / 𝑥) ∈ ℝ⁺)
11	10	rpxrd 11873	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (1 / 𝑥) ∈ ℝ^*)
12	1	blopn 22305	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ (1 / 𝑥) ∈ ℝ^*) → (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
13	4, 7, 11, 12	syl3anc 1326	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
14	13	ralrimivva 2971	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ∀𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
15		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) = (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))
16	15	fmpt2 7237	. . . . 5 ⊢ (∀𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽 ↔ (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)⟶𝐽)
17	14, 16	sylib 208	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)⟶𝐽)
18		frn 6053	. . . 4 ⊢ ((𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)⟶𝐽 → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ⊆ 𝐽)
19	17, 18	syl 17	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ⊆ 𝐽)
20		simpll 790	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝐷 ∈ (∞Met‘𝑋))
21		simprl 794	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝑢 ∈ 𝐽)
22		simprr 796	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝑧 ∈ 𝑢)
23	1	mopni2 22298	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢) → ∃𝑟 ∈ ℝ⁺ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
24	20, 21, 22, 23	syl3anc 1326	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → ∃𝑟 ∈ ℝ⁺ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
25		simprl 794	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ⁺)
26	25	rphalfcld 11884	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) → (𝑟 / 2) ∈ ℝ⁺)
27		elrp 11834	. . . . . . . 8 ⊢ ((𝑟 / 2) ∈ ℝ⁺ ↔ ((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)))
28		nnrecl 11290	. . . . . . . 8 ⊢ (((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)) → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
29	27, 28	sylbi 207	. . . . . . 7 ⊢ ((𝑟 / 2) ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
30	26, 29	syl 17	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
31	3	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝐽 ∈ Top)
32		simpr1 1067	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ⊆ 𝑋)
33	32	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝐴 ⊆ 𝑋)
34	1	mopnuni 22246	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
35	34	ad3antrrr 766	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑋 = ∪ 𝐽)
36	33, 35	sseqtrd 3641	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝐴 ⊆ ∪ 𝐽)
37		simplrr 801	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧 ∈ 𝑢)
38		simplrl 800	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑢 ∈ 𝐽)
39		elunii 4441	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝐽) → 𝑧 ∈ ∪ 𝐽)
40	37, 38, 39	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧 ∈ ∪ 𝐽)
41	40, 35	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧 ∈ 𝑋)
42		simpr3 1069	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) = 𝑋)
43	42	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ((cls‘𝐽)‘𝐴) = 𝑋)
44	41, 43	eleqtrrd 2704	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧 ∈ ((cls‘𝐽)‘𝐴))
45	20	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝐷 ∈ (∞Met‘𝑋))
46		simprrl 804	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑛 ∈ ℕ)
47	46	nnrpd 11870	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑛 ∈ ℝ⁺)
48	47	rpreccld 11882	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (1 / 𝑛) ∈ ℝ⁺)
49	48	rpxrd 11873	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (1 / 𝑛) ∈ ℝ^*)
50	1	blopn 22305	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ^*) → (𝑧(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
51	45, 41, 49, 50	syl3anc 1326	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (𝑧(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
52		blcntr 22218	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ⁺) → 𝑧 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))
53	45, 41, 48, 52	syl3anc 1326	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))
54		eqid 2622	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
55	54	clsndisj 20879	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽 ∧ 𝑧 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))) → ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ≠ ∅)
56	31, 36, 44, 51, 53, 55	syl32anc 1334	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ≠ ∅)
57		n0 3931	. . . . . . . . 9 ⊢ (((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴))
58	56, 57	sylib 208	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ∃𝑡 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴))
59	46	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑛 ∈ ℕ)
60		inss2 3834	. . . . . . . . . . . 12 ⊢ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ⊆ 𝐴
61		simpr 477	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴))
62	60, 61	sseldi 3601	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑡 ∈ 𝐴)
63		eqidd 2623	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛)))
64		oveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → (1 / 𝑥) = (1 / 𝑛))
65	64	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (𝑦(ball‘𝐷)(1 / 𝑥)) = (𝑦(ball‘𝐷)(1 / 𝑛)))
66	65	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → ((𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥)) ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑛))))
67		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → (𝑦(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛)))
68	67	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → ((𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑛)) ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛))))
69	66, 68	rspc2ev 3324	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ 𝐴 ∧ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛))) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ 𝐴 (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥)))
70	59, 62, 63, 69	syl3anc 1326	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ 𝐴 (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥)))
71		ovex 6678	. . . . . . . . . . 11 ⊢ (𝑡(ball‘𝐷)(1 / 𝑛)) ∈ V
72		eqeq1 2626	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑧 = (𝑦(ball‘𝐷)(1 / 𝑥)) ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥))))
73	72	2rexbidv 3057	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (∃𝑥 ∈ ℕ ∃𝑦 ∈ 𝐴 𝑧 = (𝑦(ball‘𝐷)(1 / 𝑥)) ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ 𝐴 (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥))))
74	15	rnmpt2 6770	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) = {𝑧 ∣ ∃𝑥 ∈ ℕ ∃𝑦 ∈ 𝐴 𝑧 = (𝑦(ball‘𝐷)(1 / 𝑥))}
75	71, 73, 74	elab2 3354	. . . . . . . . . 10 ⊢ ((𝑡(ball‘𝐷)(1 / 𝑛)) ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ 𝐴 (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥)))
76	70, 75	sylibr 224	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑡(ball‘𝐷)(1 / 𝑛)) ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))))
77		inss1 3833	. . . . . . . . . . 11 ⊢ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ⊆ (𝑧(ball‘𝐷)(1 / 𝑛))
78	77, 61	sseldi 3601	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑡 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))
79	45	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
80	49	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (1 / 𝑛) ∈ ℝ^*)
81	41	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑧 ∈ 𝑋)
82	33	adantr 481	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝐴 ⊆ 𝑋)
83	82, 62	sseldd 3604	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑡 ∈ 𝑋)
84		blcom 22199	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (1 / 𝑛) ∈ ℝ^*) ∧ (𝑧 ∈ 𝑋 ∧ 𝑡 ∈ 𝑋)) → (𝑡 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)) ↔ 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛))))
85	79, 80, 81, 83, 84	syl22anc 1327	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑡 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)) ↔ 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛))))
86	78, 85	mpbid 222	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛)))
87		simprll 802	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑟 ∈ ℝ⁺)
88	87	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑟 ∈ ℝ⁺)
89	88	rphalfcld 11884	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑟 / 2) ∈ ℝ⁺)
90	89	rpxrd 11873	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑟 / 2) ∈ ℝ^*)
91		simprrr 805	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (1 / 𝑛) < (𝑟 / 2))
92	87	rphalfcld 11884	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (𝑟 / 2) ∈ ℝ⁺)
93		rpre 11839	. . . . . . . . . . . . . . 15 ⊢ ((1 / 𝑛) ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ)
94		rpre 11839	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 / 2) ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ)
95		ltle 10126	. . . . . . . . . . . . . . 15 ⊢ (((1 / 𝑛) ∈ ℝ ∧ (𝑟 / 2) ∈ ℝ) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
96	93, 94, 95	syl2an 494	. . . . . . . . . . . . . 14 ⊢ (((1 / 𝑛) ∈ ℝ⁺ ∧ (𝑟 / 2) ∈ ℝ⁺) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
97	48, 92, 96	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
98	91, 97	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (1 / 𝑛) ≤ (𝑟 / 2))
99	98	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (1 / 𝑛) ≤ (𝑟 / 2))
100		ssbl 22228	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋) ∧ ((1 / 𝑛) ∈ ℝ^* ∧ (𝑟 / 2) ∈ ℝ^*) ∧ (1 / 𝑛) ≤ (𝑟 / 2)) → (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ (𝑡(ball‘𝐷)(𝑟 / 2)))
101	79, 83, 80, 90, 99, 100	syl221anc 1337	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ (𝑡(ball‘𝐷)(𝑟 / 2)))
102	88	rpred 11872	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑟 ∈ ℝ)
103	101, 86	sseldd 3604	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑧 ∈ (𝑡(ball‘𝐷)(𝑟 / 2)))
104		blhalf 22210	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋) ∧ (𝑟 ∈ ℝ ∧ 𝑧 ∈ (𝑡(ball‘𝐷)(𝑟 / 2)))) → (𝑡(ball‘𝐷)(𝑟 / 2)) ⊆ (𝑧(ball‘𝐷)𝑟))
105	79, 83, 102, 103, 104	syl22anc 1327	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑡(ball‘𝐷)(𝑟 / 2)) ⊆ (𝑧(ball‘𝐷)𝑟))
106		simprlr 803	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
107	106	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
108	105, 107	sstrd 3613	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑡(ball‘𝐷)(𝑟 / 2)) ⊆ 𝑢)
109	101, 108	sstrd 3613	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢)
110		eleq2 2690	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛))))
111		sseq1 3626	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑤 ⊆ 𝑢 ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢))
112	110, 111	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢) ↔ (𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛)) ∧ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢)))
113	112	rspcev 3309	. . . . . . . . 9 ⊢ (((𝑡(ball‘𝐷)(1 / 𝑛)) ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∧ (𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛)) ∧ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢)) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢))
114	76, 86, 109, 113	syl12anc 1324	. . . . . . . 8 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢))
115	58, 114	exlimddv 1863	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ ((𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢))
116	115	anassrs 680	. . . . . 6 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2))) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢))
117	30, 116	rexlimddv 3035	. . . . 5 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢))
118	24, 117	rexlimddv 3035	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢))
119	118	ralrimivva 2971	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ∀𝑢 ∈ 𝐽 ∀𝑧 ∈ 𝑢 ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢))
120		basgen2 20793	. . 3 ⊢ ((𝐽 ∈ Top ∧ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ⊆ 𝐽 ∧ ∀𝑢 ∈ 𝐽 ∀𝑧 ∈ 𝑢 ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) = 𝐽)
121	3, 19, 119, 120	syl3anc 1326	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) = 𝐽)
122	121, 3	eqeltrd 2701	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ Top)
123		tgclb 20774	. . . 4 ⊢ (ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∈ TopBases ↔ (topGen‘ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ Top)
124	122, 123	sylibr 224	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∈ TopBases)
125		omelon 8543	. . . . . 6 ⊢ ω ∈ On
126		simpr2 1068	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ≼ ω)
127		nnex 11026	. . . . . . . . 9 ⊢ ℕ ∈ V
128	127	xpdom2 8055	. . . . . . . 8 ⊢ (𝐴 ≼ ω → (ℕ × 𝐴) ≼ (ℕ × ω))
129	126, 128	syl 17	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ≼ (ℕ × ω))
130		nnenom 12779	. . . . . . . . 9 ⊢ ℕ ≈ ω
131		omex 8540	. . . . . . . . . 10 ⊢ ω ∈ V
132	131	enref 7988	. . . . . . . . 9 ⊢ ω ≈ ω
133		xpen 8123	. . . . . . . . 9 ⊢ ((ℕ ≈ ω ∧ ω ≈ ω) → (ℕ × ω) ≈ (ω × ω))
134	130, 132, 133	mp2an 708	. . . . . . . 8 ⊢ (ℕ × ω) ≈ (ω × ω)
135		xpomen 8838	. . . . . . . 8 ⊢ (ω × ω) ≈ ω
136	134, 135	entri 8010	. . . . . . 7 ⊢ (ℕ × ω) ≈ ω
137		domentr 8015	. . . . . . 7 ⊢ (((ℕ × 𝐴) ≼ (ℕ × ω) ∧ (ℕ × ω) ≈ ω) → (ℕ × 𝐴) ≼ ω)
138	129, 136, 137	sylancl 694	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ≼ ω)
139		ondomen 8860	. . . . . 6 ⊢ ((ω ∈ On ∧ (ℕ × 𝐴) ≼ ω) → (ℕ × 𝐴) ∈ dom card)
140	125, 138, 139	sylancr 695	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ∈ dom card)
141		ffn 6045	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)⟶𝐽 → (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) Fn (ℕ × 𝐴))
142	17, 141	syl 17	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) Fn (ℕ × 𝐴))
143		dffn4 6121	. . . . . 6 ⊢ ((𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) Fn (ℕ × 𝐴) ↔ (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)–onto→ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))))
144	142, 143	sylib 208	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)–onto→ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))))
145		fodomnum 8880	. . . . 5 ⊢ ((ℕ × 𝐴) ∈ dom card → ((𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)–onto→ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ (ℕ × 𝐴)))
146	140, 144, 145	sylc 65	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ (ℕ × 𝐴))
147		domtr 8009	. . . 4 ⊢ ((ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ (ℕ × 𝐴) ∧ (ℕ × 𝐴) ≼ ω) → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ ω)
148	146, 138, 147	syl2anc 693	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ ω)
149		2ndci 21251	. . 3 ⊢ ((ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∈ TopBases ∧ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ ω) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ 2^nd𝜔)
150	124, 148, 149	syl2anc 693	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ 2^nd𝜔)
151	121, 150	eqeltrrd 2702	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ 2^nd𝜔)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 class class class wbr 4653 × cxp 5112 dom cdm 5114 ran crn 5115 Oncon0 5723 Fn wfn 5883 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ωcom 7065 ≈ cen 7952 ≼ cdom 7953 cardccrd 8761 ℝcr 9935 0cc0 9936 1c1 9937 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 / cdiv 10684 ℕcn 11020 2c2 11070 ℝ⁺crp 11832 topGenctg 16098 ∞Metcxmt 19731 ballcbl 19733 MetOpencmopn 19736 Topctop 20698 TopBasesctb 20749 clsccl 20822 2^nd𝜔c2ndc 21241

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-2ndc 21243

This theorem is referenced by: met2ndc 22328

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