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Theorem met2ndci 22825

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 22743	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
3	2	adantr 473	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ Top)
4		simpll 754	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
5		simplr1 1195	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝐴 ⊆ 𝑋)
6		simprr 760	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
7	5, 6	sseldd 3855	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝑋)
8		simprl 758	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ)
9	8	nnrpd 12239	. . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ⁺)
10	9	rpreccld 12251	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (1 / 𝑥) ∈ ℝ⁺)
11	10	rpxrd 12242	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (1 / 𝑥) ∈ ℝ^*)
12	1	blopn 22803	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ (1 / 𝑥) ∈ ℝ^*) → (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
13	4, 7, 11, 12	syl3anc 1351	. . . . . 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 3135	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ∀𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
15		eqid 2772	. . . . . 6 ⊢ (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) = (𝑥 ∈ ℕ, 𝑦 ∈ 𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))
16	15	fmpo 7567	. . . . 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 210	. . . 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 6345	. . 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 754	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝐷 ∈ (∞Met‘𝑋))
20		simprl 758	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝑢 ∈ 𝐽)
21		simprr 760	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢)) → 𝑧 ∈ 𝑢)
22	1	mopni2 22796	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢) → ∃𝑟 ∈ ℝ⁺ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
23	19, 20, 21, 22	syl3anc 1351	. . . . 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 758	. . . . . . . 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 12253	. . . . . . 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 12199	. . . . . . . 8 ⊢ ((𝑟 / 2) ∈ ℝ⁺ ↔ ((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)))
27		nnrecl 11698	. . . . . . . 8 ⊢ (((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)) → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
28	26, 27	sylbi 209	. . . . . . 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 713	. . . . . . . . . 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 1174	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ⊆ 𝑋)
32	31	ad2antrr 713	. . . . . . . . . . 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 22744	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
34	33	ad3antrrr 717	. . . . . . . . . . 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 3893	. . . . . . . . . 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 765	. . . . . . . . . . . . 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 764	. . . . . . . . . . . . 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 4711	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝐽) → 𝑧 ∈ ∪ 𝐽)
39	36, 37, 38	syl2anc 576	. . . . . . . . . . . 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 2863	. . . . . . . . . . 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 1176	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) = 𝑋)
42	41	ad2antrr 713	. . . . . . . . . . 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 2863	. . . . . . . . . 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 473	. . . . . . . . . . 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 768	. . . . . . . . . . . . . 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 12239	. . . . . . . . . . . . 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 12251	. . . . . . . . . . . 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 12242	. . . . . . . . . . 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 22803	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ^*) → (𝑧(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
50	44, 40, 48, 49	syl3anc 1351	. . . . . . . . . 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 22716	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ⁺) → 𝑧 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))
52	44, 40, 47, 51	syl3anc 1351	. . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
54	53	clsndisj 21377	. . . . . . . . . 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 1358	. . . . . . . . 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 4191	. . . . . . . . 9 ⊢ (((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴))
57	55, 56	sylib 210	. . . . . . . 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 473	. . . . . . . . . . 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 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))
60	59	elin2d 4060	. . . . . . . . . . 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 2773	. . . . . . . . . . 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 6978	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → (1 / 𝑥) = (1 / 𝑛))
63	62	oveq2d 6986	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (𝑦(ball‘𝐷)(1 / 𝑥)) = (𝑦(ball‘𝐷)(1 / 𝑛)))
64	63	eqeq2d 2782	. . . . . . . . . . . 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 6977	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → (𝑦(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛)))
66	65	eqeq2d 2782	. . . . . . . . . . . 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 3544	. . . . . . . . . . 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 1351	. . . . . . . . . 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 7002	. . . . . . . . . . 11 ⊢ (𝑡(ball‘𝐷)(1 / 𝑛)) ∈ V
70		eqeq1 2776	. . . . . . . . . . . 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 3239	. . . . . . . . . . 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 7094	. . . . . . . . . . 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 3579	. . . . . . . . . 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 226	. . . . . . . . 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 4059	. . . . . . . . . 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 473	. . . . . . . . . . 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 473	. . . . . . . . . . 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 473	. . . . . . . . . . 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 473	. . . . . . . . . . . 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 3855	. . . . . . . . . . 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 22697	. . . . . . . . . . 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 826	. . . . . . . . . 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 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)) → 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛)))
84		simprll 766	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . 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 12253	. . . . . . . . . . . 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 12242	. . . . . . . . . . 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 769	. . . . . . . . . . . . 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 12253	. . . . . . . . . . . . . 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 12205	. . . . . . . . . . . . . . 15 ⊢ ((1 / 𝑛) ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ)
91		rpre 12205	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 / 2) ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ)
92		ltle 10521	. . . . . . . . . . . . . . 15 ⊢ (((1 / 𝑛) ∈ ℝ ∧ (𝑟 / 2) ∈ ℝ) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
93	90, 91, 92	syl2an 586	. . . . . . . . . . . . . 14 ⊢ (((1 / 𝑛) ∈ ℝ⁺ ∧ (𝑟 / 2) ∈ ℝ⁺) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
94	47, 89, 93	syl2anc 576	. . . . . . . . . . . . 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 473	. . . . . . . . . . 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 22726	. . . . . . . . . . 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 1361	. . . . . . . . . 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 12241	. . . . . . . . . . . 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 3855	. . . . . . . . . . . 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 22708	. . . . . . . . . . . 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 826	. . . . . . . . . . 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 767	. . . . . . . . . . . 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 473	. . . . . . . . . . 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 3864	. . . . . . . . . 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 3864	. . . . . . . . 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 2848	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛))))
108		sseq1 3878	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑤 ⊆ 𝑢 ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢))
109	107, 108	anbi12d 621	. . . . . . . . . 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 3529	. . . . . . . . 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 824	. . . . . . . 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 1894	. . . . . . 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 460	. . . . . 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 3230	. . . . 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 3230	. . . 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 3135	. . 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 21291	. . 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 1351	. 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 2860	. . . 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 21272	. . . 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 226	. . 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 8895	. . . . . 6 ⊢ ω ∈ On
123		simpr2 1175	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ≼ ω)
124		nnex 11438	. . . . . . . . 9 ⊢ ℕ ∈ V
125	124	xpdom2 8400	. . . . . . . 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 13156	. . . . . . . . 9 ⊢ ℕ ≈ ω
128		omex 8892	. . . . . . . . . 10 ⊢ ω ∈ V
129	128	enref 8331	. . . . . . . . 9 ⊢ ω ≈ ω
130		xpen 8468	. . . . . . . . 9 ⊢ ((ℕ ≈ ω ∧ ω ≈ ω) → (ℕ × ω) ≈ (ω × ω))
131	127, 129, 130	mp2an 679	. . . . . . . 8 ⊢ (ℕ × ω) ≈ (ω × ω)
132		xpomen 9227	. . . . . . . 8 ⊢ (ω × ω) ≈ ω
133	131, 132	entri 8352	. . . . . . 7 ⊢ (ℕ × ω) ≈ ω
134		domentr 8357	. . . . . . 7 ⊢ (((ℕ × 𝐴) ≼ (ℕ × ω) ∧ (ℕ × ω) ≈ ω) → (ℕ × 𝐴) ≼ ω)
135	126, 133, 134	sylancl 577	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ≼ ω)
136		ondomen 9249	. . . . . 6 ⊢ ((ω ∈ On ∧ (ℕ × 𝐴) ≼ ω) → (ℕ × 𝐴) ∈ dom card)
137	122, 135, 136	sylancr 578	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ∈ dom card)
138	17	ffnd 6339	. . . . . 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 6419	. . . . . 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 210	. . . . 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 9269	. . . . 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 8351	. . . 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 576	. . 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 21750	. . 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 576	. 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 2861	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 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∃wex 1742 ∈ wcel 2048 ≠ wne 2961 ∀wral 3082 ∃wrex 3083 ∩ cin 3824 ⊆ wss 3825 ∅c0 4173 ∪ cuni 4706 class class class wbr 4923 × cxp 5398 dom cdm 5400 ran crn 5401 Oncon0 6023 Fn wfn 6177 ⟶wf 6178 –onto→wfo 6180 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 ωcom 7390 ≈ cen 8295 ≼ cdom 8296 cardccrd 9150 ℝcr 10326 0cc0 10327 1c1 10328 ℝ^*cxr 10465 < clt 10466 ≤ cle 10467 / cdiv 11090 ℕcn 11431 2c2 11488 ℝ⁺crp 12197 topGenctg 16557 ∞Metcxmet 20222 ballcbl 20224 MetOpencmopn 20227 Topctop 21195 TopBasesctb 21247 clsccl 21320 2^nd𝜔c2ndc 21740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-inf 8694 df-oi 8761 df-card 9154 df-acn 9157 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-n0 11701 df-z 11787 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-topgen 16563 df-psmet 20229 df-xmet 20230 df-bl 20232 df-mopn 20233 df-top 21196 df-topon 21213 df-bases 21248 df-cld 21321 df-ntr 21322 df-cls 21323 df-2ndc 21742

This theorem is referenced by: met2ndc 22826

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