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Theorem met2ndci 24589

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 24507	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
3	2	adantr 484	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ Top)
4		simpll 776	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
5		simplr1 1230	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝐴 ⊆ 𝑋)
6		simprr 782	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
7	5, 6	sseldd 3938	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝑋)
8		simprl 780	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ)
9	8	nnrpd 13045	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ⁺)
10	9	rpreccld 13057	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (1 / 𝑥) ∈ ℝ⁺)
11	10	rpxrd 13048	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (1 / 𝑥) ∈ ℝ^*)
12	1	blopn 24567	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ (1 / 𝑥) ∈ ℝ^*) → (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
13	4, 7, 11, 12	syl3anc 1392	. . . . . 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 3206	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ∀𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
15		eqid 2763	. . . . . 6 ⊢ (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) = (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))
16	15	fmpo 8049	. . . . 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 220	. . . 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	17	frnd 6700	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ⊆ 𝐽)
19		simpll 776	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝐷 ∈ (∞Met‘𝑋))
20		simprl 780	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝑢 ∈ 𝐽)
21		simprr 782	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝑧 ∈ 𝑢)
22	1	mopni2 24560	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢) → ∃𝑟 ∈ ℝ⁺ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
23	19, 20, 21, 22	syl3anc 1392	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → ∃𝑟 ∈ ℝ⁺ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
24		simprl 780	. . . . . . . 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⁺)
25	24	rphalfcld 13059	. . . . . . 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⁺)
26		elrp 13005	. . . . . . . 8 ⊢ ((𝑟 / 2) ∈ ℝ⁺ ↔ ((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)))
27		nnrecl 12489	. . . . . . . 8 ⊢ (((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)) → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
28	26, 27	sylbi 219	. . . . . . 7 ⊢ ((𝑟 / 2) ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
29	25, 28	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))
30	3	ad2antrr 736	. . . . . . . . . 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)
31		simpr1 1209	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ⊆ 𝑋)
32	31	ad2antrr 736	. . . . . . . . . . 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)
33	1	mopnuni 24508	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
34	33	ad3antrrr 740	. . . . . . . . . . 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)
35	32, 34	sseqtrd 3973	. . . . . . . . . 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)
36		simplrr 787	. . . . . . . . . . . . 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)
37		simplrl 786	. . . . . . . . . . . . 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)
38		elunii 4871	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝐽) → 𝑧 ∈ ∪ 𝐽)
39	36, 37, 38	syl2anc 593	. . . . . . . . . . . 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)
40	39, 34	eleqtrrd 2866	. . . . . . . . . . 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)
41		simpr3 1211	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) = 𝑋)
42	41	ad2antrr 736	. . . . . . . . . . 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)
43	40, 42	eleqtrrd 2866	. . . . . . . . . 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))
44	19	adantr 484	. . . . . . . . . . 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))
45		simprrl 790	. . . . . . . . . . . . . 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)
46	45	nnrpd 13045	. . . . . . . . . . . . 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⁺)
47	46	rpreccld 13057	. . . . . . . . . . . 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⁺)
48	47	rpxrd 13048	. . . . . . . . . . 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^*)
49	1	blopn 24567	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ^*) → (𝑧(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
50	44, 40, 48, 49	syl3anc 1392	. . . . . . . . . 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)
51		blcntr 24480	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ⁺) → 𝑧 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))
52	44, 40, 47, 51	syl3anc 1392	. . . . . . . . . 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)))
53		eqid 2763	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
54	53	clsndisj 23142	. . . . . . . . . 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) ≠ ∅)
55	30, 35, 43, 50, 52, 54	syl32anc 1399	. . . . . . . . 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) ≠ ∅)
56		n0 4306	. . . . . . . . 9 ⊢ (((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴))
57	55, 56	sylib 220	. . . . . . . 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))
58	45	adantr 484	. . . . . . . . . . 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)
59		simpr 488	. . . . . . . . . . . 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))
60	59	elin2d 4158	. . . . . . . . . . 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)
61		eqidd 2764	. . . . . . . . . . 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)))
62		oveq2 7404	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → (1 / 𝑥) = (1 / 𝑛))
63	62	oveq2d 7412	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (𝑦(ball‘𝐷)(1 / 𝑥)) = (𝑦(ball‘𝐷)(1 / 𝑛)))
64	63	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → ((𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥)) ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑛))))
65		oveq1 7403	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → (𝑦(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛)))
66	65	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → ((𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑛)) ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛))))
67	64, 66	rspc2ev 3595	. . . . . . . . . . 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)))
68	58, 60, 61, 67	syl3anc 1392	. . . . . . . . . 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)))
69		ovex 7429	. . . . . . . . . . 11 ⊢ (𝑡(ball‘𝐷)(1 / 𝑛)) ∈ V
70		eqeq1 2767	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑧 = (𝑦(ball‘𝐷)(1 / 𝑥)) ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥))))
71	70	2rexbidv 3228	. . . . . . . . . . 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))))
72	15	rnmpo 7529	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) = {𝑧 ∣ ∃𝑥 ∈ ℕ ∃𝑦 ∈ 𝐴 𝑧 = (𝑦(ball‘𝐷)(1 / 𝑥))}
73	69, 71, 72	elab2 3642	. . . . . . . . . 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)))
74	68, 73	sylibr 236	. . . . . . . . 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))))
75	59	elin1d 4157	. . . . . . . . . 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)))
76	44	adantr 484	. . . . . . . . . . 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))
77	48	adantr 484	. . . . . . . . . . 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^*)
78	40	adantr 484	. . . . . . . . . . 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)
79	32	adantr 484	. . . . . . . . . . . 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)
80	79, 60	sseldd 3938	. . . . . . . . . . 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)
81		blcom 24461	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (1 / 𝑛) ∈ ℝ^*) ∧ (𝑧 ∈ 𝑋 ∧ 𝑡 ∈ 𝑋)) → (𝑡 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)) ↔ 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛))))
82	76, 77, 78, 80, 81	syl22anc 849	. . . . . . . . . 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))))
83	75, 82	mpbid 234	. . . . . . . . 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)))
84		simprll 788	. . . . . . . . . . . . . 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⁺)
85	84	adantr 484	. . . . . . . . . . . . 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⁺)
86	85	rphalfcld 13059	. . . . . . . . . . . 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⁺)
87	86	rpxrd 13048	. . . . . . . . . . 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^*)
88		simprrr 791	. . . . . . . . . . . . 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))
89	84	rphalfcld 13059	. . . . . . . . . . . . . 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⁺)
90		rpre 13012	. . . . . . . . . . . . . . 15 ⊢ ((1 / 𝑛) ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ)
91		rpre 13012	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 / 2) ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ)
92		ltle 11282	. . . . . . . . . . . . . . 15 ⊢ (((1 / 𝑛) ∈ ℝ ∧ (𝑟 / 2) ∈ ℝ) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
93	90, 91, 92	syl2an 605	. . . . . . . . . . . . . 14 ⊢ (((1 / 𝑛) ∈ ℝ⁺ ∧ (𝑟 / 2) ∈ ℝ⁺) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
94	47, 89, 93	syl2anc 593	. . . . . . . . . . . . 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)))
95	88, 94	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))
96	95	adantr 484	. . . . . . . . . . 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))
97		ssbl 24490	. . . . . . . . . . 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)))
98	76, 80, 77, 87, 96, 97	syl221anc 1402	. . . . . . . . . 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)))
99	85	rpred 13047	. . . . . . . . . . . 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)
100	98, 83	sseldd 3938	. . . . . . . . . . . 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)))
101		blhalf 24472	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋) ∧ (𝑟 ∈ ℝ ∧ 𝑧 ∈ (𝑡(ball‘𝐷)(𝑟 / 2)))) → (𝑡(ball‘𝐷)(𝑟 / 2)) ⊆ (𝑧(ball‘𝐷)𝑟))
102	76, 80, 99, 100, 101	syl22anc 849	. . . . . . . . . . 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))
103		simprlr 789	. . . . . . . . . . . 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)
104	103	adantr 484	. . . . . . . . . . 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)
105	102, 104	sstrd 3947	. . . . . . . . . 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)
106	98, 105	sstrd 3947	. . . . . . . . 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)
107		eleq2 2852	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛))))
108		sseq1 3962	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑤 ⊆ 𝑢 ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢))
109	107, 108	anbi12d 641	. . . . . . . . . 10 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢) ↔ (𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛)) ∧ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢)))
110	109	rspcev 3582	. . . . . . . . 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))
111	74, 83, 106, 110	syl12anc 847	. . . . . . . 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))
112	57, 111	exlimddv 1956	. . . . . . 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))
113	112	anassrs 471	. . . . . 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))
114	29, 113	rexlimddv 3170	. . . . 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))
115	23, 114	rexlimddv 3170	. . . 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))
116	115	ralrimivva 3206	. . 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))
117		basgen2 23056	. . 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)
118	3, 18, 116, 117	syl3anc 1392	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) = 𝐽)
119	118, 3	eqeltrd 2863	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ Top)
120		tgclb 23037	. . . 4 ⊢ (ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∈ TopBases ↔ (topGen‘ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ Top)
121	119, 120	sylibr 236	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∈ TopBases)
122		omelon 9599	. . . . . 6 ⊢ ω ∈ On
123		simpr2 1210	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ≼ ω)
124		nnex 12226	. . . . . . . . 9 ⊢ ℕ ∈ V
125	124	xpdom2 9044	. . . . . . . 8 ⊢ (𝐴 ≼ ω → (ℕ × 𝐴) ≼ (ℕ × ω))
126	123, 125	syl 17	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ≼ (ℕ × ω))
127		nnenom 14003	. . . . . . . . 9 ⊢ ℕ ≈ ω
128		omex 9596	. . . . . . . . . 10 ⊢ ω ∈ V
129	128	enref 8966	. . . . . . . . 9 ⊢ ω ≈ ω
130		xpen 9112	. . . . . . . . 9 ⊢ ((ℕ ≈ ω ∧ ω ≈ ω) → (ℕ × ω) ≈ (ω × ω))
131	127, 129, 130	mp2an 702	. . . . . . . 8 ⊢ (ℕ × ω) ≈ (ω × ω)
132		xpomen 9983	. . . . . . . 8 ⊢ (ω × ω) ≈ ω
133	131, 132	entri 8989	. . . . . . 7 ⊢ (ℕ × ω) ≈ ω
134		domentr 8994	. . . . . . 7 ⊢ (((ℕ × 𝐴) ≼ (ℕ × ω) ∧ (ℕ × ω) ≈ ω) → (ℕ × 𝐴) ≼ ω)
135	126, 133, 134	sylancl 595	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ≼ ω)
136		ondomen 10005	. . . . . 6 ⊢ ((ω ∈ On ∧ (ℕ × 𝐴) ≼ ω) → (ℕ × 𝐴) ∈ dom card)
137	122, 135, 136	sylancr 596	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ∈ dom card)
138	17	ffnd 6692	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) Fn (ℕ × 𝐴))
139		dffn4 6784	. . . . . 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))))
140	138, 139	sylib 220	. . . . 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))))
141		fodomnum 10025	. . . . 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)))
142	137, 140, 141	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))
143		domtr 8988	. . . 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)
144	142, 135, 143	syl2anc 593	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ ω)
145		2ndci 23515	. . 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)
146	121, 144, 145	syl2anc 593	. 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)
147	118, 146	eqeltrrd 2864	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 208 ∧ wa 399 ∧ w3a 1099 = wceq 1561 ∃wex 1800 ∈ wcel 2143 ≠ wne 2958 ∀wral 3077 ∃wrex 3087 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 ∪ cuni 4866 class class class wbr 5101 × cxp 5646 dom cdm 5648 ran crn 5649 Oncon0 6346 Fn wfn 6516 ⟶wf 6517 –onto→wfo 6519 ‘cfv 6521 (class class class)co 7396 ∈ cmpo 7398 ωcom 7846 ≈ cen 8924 ≼ cdom 8925 cardccrd 9905 ℝcr 11083 0cc0 11084 1c1 11085 ℝ^*cxr 11226 < clt 11227 ≤ cle 11228 / cdiv 11855 ℕcn 12220 2c2 12282 ℝ⁺crp 13003 topGenctg 17476 ∞Metcxmet 21416 ballcbl 21418 MetOpencmopn 21421 Topctop 22960 TopBasesctb 23012 clsccl 23085 2^ndωc2ndc 23505

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-inf2 9594 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161 ax-pre-sup 11162

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9386 df-inf 9387 df-oi 9456 df-card 9909 df-acn 9912 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-div 11856 df-nn 12221 df-2 12290 df-n0 12492 df-z 12579 df-uz 12850 df-q 12960 df-rp 13004 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-topgen 17482 df-psmet 21423 df-xmet 21424 df-bl 21426 df-mopn 21427 df-top 22961 df-topon 22978 df-bases 23013 df-cld 23086 df-ntr 23087 df-cls 23088 df-2ndc 23507

This theorem is referenced by: met2ndc 24590

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