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Theorem met2ndci 22137

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 22055	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
3	2	adantr 480	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ Top)
4		simpll 786	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
5		simplr1 1096	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝐴 ⊆ 𝑋)
6		simprr 792	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
7	5, 6	sseldd 3569	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝑋)
8		simprl 790	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ)
9	8	nnrpd 11746	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ⁺)
10	9	rpreccld 11758	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (1 / 𝑥) ∈ ℝ⁺)
11	10	rpxrd 11749	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (1 / 𝑥) ∈ ℝ^*)
12	1	blopn 22115	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ (1 / 𝑥) ∈ ℝ^*) → (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
13	4, 7, 11, 12	syl3anc 1318	. . . . . 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 2954	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ∀𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
15		eqid 2610	. . . . . 6 ⊢ (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) = (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))
16	15	fmpt2 7126	. . . . 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 207	. . . 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 5966	. . . 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 786	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝐷 ∈ (∞Met‘𝑋))
21		simprl 790	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝑢 ∈ 𝐽)
22		simprr 792	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝑧 ∈ 𝑢)
23	1	mopni2 22108	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢) → ∃𝑟 ∈ ℝ⁺ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
24	20, 21, 22, 23	syl3anc 1318	. . . . 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 790	. . . . . . . 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 11760	. . . . . . 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 11710	. . . . . . . 8 ⊢ ((𝑟 / 2) ∈ ℝ⁺ ↔ ((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)))
28		nnrecl 11167	. . . . . . . 8 ⊢ (((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)) → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
29	27, 28	sylbi 206	. . . . . . 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 758	. . . . . . . . . 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 1060	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ⊆ 𝑋)
33	32	ad2antrr 758	. . . . . . . . . . 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 22056	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
35	34	ad3antrrr 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)))) → 𝑋 = ∪ 𝐽)
36	33, 35	sseqtrd 3604	. . . . . . . . . 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 797	. . . . . . . . . . . . 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 796	. . . . . . . . . . . . 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 4377	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝐽) → 𝑧 ∈ ∪ 𝐽)
40	37, 38, 39	syl2anc 691	. . . . . . . . . . . 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 2691	. . . . . . . . . . 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 1062	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) = 𝑋)
43	42	ad2antrr 758	. . . . . . . . . . 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 2691	. . . . . . . . . 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 480	. . . . . . . . . . 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 800	. . . . . . . . . . . . . 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 11746	. . . . . . . . . . . . 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 11758	. . . . . . . . . . . 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 11749	. . . . . . . . . . 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 22115	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ^*) → (𝑧(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
51	45, 41, 49, 50	syl3anc 1318	. . . . . . . . . 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 22028	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ⁺) → 𝑧 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))
53	45, 41, 48, 52	syl3anc 1318	. . . . . . . . . 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 2610	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
55	54	clsndisj 20689	. . . . . . . . . 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 1326	. . . . . . . . 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 3890	. . . . . . . . 9 ⊢ (((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴))
58	56, 57	sylib 207	. . . . . . . 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 480	. . . . . . . . . . 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 3796	. . . . . . . . . . . 12 ⊢ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ⊆ 𝐴
61		simpr 476	. . . . . . . . . . . 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 3566	. . . . . . . . . . 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 2611	. . . . . . . . . . 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 6557	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → (1 / 𝑥) = (1 / 𝑛))
65	64	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (𝑦(ball‘𝐷)(1 / 𝑥)) = (𝑦(ball‘𝐷)(1 / 𝑛)))
66	65	eqeq2d 2620	. . . . . . . . . . . 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 6556	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → (𝑦(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛)))
68	67	eqeq2d 2620	. . . . . . . . . . . 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 3295	. . . . . . . . . . 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 1318	. . . . . . . . . 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 6577	. . . . . . . . . . 11 ⊢ (𝑡(ball‘𝐷)(1 / 𝑛)) ∈ V
72		eqeq1 2614	. . . . . . . . . . . 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 3039	. . . . . . . . . . 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 6668	. . . . . . . . . . 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 3323	. . . . . . . . . 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 223	. . . . . . . . 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 3795	. . . . . . . . . . 11 ⊢ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ⊆ (𝑧(ball‘𝐷)(1 / 𝑛))
78	77, 61	sseldi 3566	. . . . . . . . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . . . . 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 3569	. . . . . . . . . . 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 22009	. . . . . . . . . . 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 1319	. . . . . . . . . 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 221	. . . . . . . . 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 798	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 11760	. . . . . . . . . . . 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 11749	. . . . . . . . . . 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 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)))) → (1 / 𝑛) < (𝑟 / 2))
92	87	rphalfcld 11760	. . . . . . . . . . . . . 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 11715	. . . . . . . . . . . . . . 15 ⊢ ((1 / 𝑛) ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ)
94		rpre 11715	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 / 2) ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ)
95		ltle 10005	. . . . . . . . . . . . . . 15 ⊢ (((1 / 𝑛) ∈ ℝ ∧ (𝑟 / 2) ∈ ℝ) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
96	93, 94, 95	syl2an 493	. . . . . . . . . . . . . 14 ⊢ (((1 / 𝑛) ∈ ℝ⁺ ∧ (𝑟 / 2) ∈ ℝ⁺) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
97	48, 92, 96	syl2anc 691	. . . . . . . . . . . . 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 480	. . . . . . . . . . 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 22038	. . . . . . . . . . 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 1329	. . . . . . . . . 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 11748	. . . . . . . . . . . 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 3569	. . . . . . . . . . . 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 22020	. . . . . . . . . . . 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 1319	. . . . . . . . . . 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 799	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 3578	. . . . . . . . . 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 3578	. . . . . . . . 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 2677	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛))))
111		sseq1 3589	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑤 ⊆ 𝑢 ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢))
112	110, 111	anbi12d 743	. . . . . . . . . 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 3282	. . . . . . . . 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 1316	. . . . . . . 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 1850	. . . . . . 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 678	. . . . . 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 3017	. . . . 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 3017	. . . 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 2954	. . 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 20604	. . 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 1318	. 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 2688	. . . 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 20585	. . . 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 223	. . 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 8426	. . . . . 6 ⊢ ω ∈ On
126		simpr2 1061	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ≼ ω)
127		nnex 10903	. . . . . . . . 9 ⊢ ℕ ∈ V
128	127	xpdom2 7940	. . . . . . . 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 12641	. . . . . . . . 9 ⊢ ℕ ≈ ω
131		omex 8423	. . . . . . . . . 10 ⊢ ω ∈ V
132	131	enref 7874	. . . . . . . . 9 ⊢ ω ≈ ω
133		xpen 8008	. . . . . . . . 9 ⊢ ((ℕ ≈ ω ∧ ω ≈ ω) → (ℕ × ω) ≈ (ω × ω))
134	130, 132, 133	mp2an 704	. . . . . . . 8 ⊢ (ℕ × ω) ≈ (ω × ω)
135		xpomen 8721	. . . . . . . 8 ⊢ (ω × ω) ≈ ω
136	134, 135	entri 7896	. . . . . . 7 ⊢ (ℕ × ω) ≈ ω
137		domentr 7901	. . . . . . 7 ⊢ (((ℕ × 𝐴) ≼ (ℕ × ω) ∧ (ℕ × ω) ≈ ω) → (ℕ × 𝐴) ≼ ω)
138	129, 136, 137	sylancl 693	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ≼ ω)
139		ondomen 8743	. . . . . 6 ⊢ ((ω ∈ On ∧ (ℕ × 𝐴) ≼ ω) → (ℕ × 𝐴) ∈ dom card)
140	125, 138, 139	sylancr 694	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ∈ dom card)
141		ffn 5958	. . . . . . 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 6034	. . . . . 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 207	. . . . 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 8763	. . . . 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 63	. . . 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 7895	. . . 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 691	. . 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 21061	. . 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 691	. 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 2689	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 class class class wbr 4583 × cxp 5036 dom cdm 5038 ran crn 5039 Oncon0 5640 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 cardccrd 8644 ℝcr 9814 0cc0 9815 1c1 9816 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 / cdiv 10563 ℕcn 10897 2c2 10947 ℝ⁺crp 11708 topGenctg 15921 ∞Metcxmt 19552 ballcbl 19554 MetOpencmopn 19557 Topctop 20517 TopBasesctb 20520 clsccl 20632 2^nd𝜔c2ndc 21051

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-2ndc 21053

This theorem is referenced by: met2ndc 22138

W3C validator