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Theorem met2ndci 22237
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) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ 2nd𝜔)

Proof of Theorem met2ndci
Dummy variables 𝑛 𝑟 𝑡 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 methaus.1 . . . . 5 𝐽 = (MetOpen‘𝐷)
21mopntop 22155 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
32adantr 481 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ Top)
4 simpll 789 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
5 simplr1 1101 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴𝑋)
6 simprr 795 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦𝐴)
75, 6sseldd 3584 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦𝑋)
8 simprl 793 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦𝐴)) → 𝑥 ∈ ℕ)
98nnrpd 11814 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦𝐴)) → 𝑥 ∈ ℝ+)
109rpreccld 11826 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦𝐴)) → (1 / 𝑥) ∈ ℝ+)
1110rpxrd 11817 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦𝐴)) → (1 / 𝑥) ∈ ℝ*)
121blopn 22215 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋 ∧ (1 / 𝑥) ∈ ℝ*) → (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
134, 7, 11, 12syl3anc 1323 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑥 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
1413ralrimivva 2965 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ∀𝑥 ∈ ℕ ∀𝑦𝐴 (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽)
15 eqid 2621 . . . . . 6 (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) = (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))
1615fmpt2 7182 . . . . 5 (∀𝑥 ∈ ℕ ∀𝑦𝐴 (𝑦(ball‘𝐷)(1 / 𝑥)) ∈ 𝐽 ↔ (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)⟶𝐽)
1714, 16sylib 208 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)⟶𝐽)
18 frn 6010 . . . 4 ((𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)⟶𝐽 → ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ⊆ 𝐽)
1917, 18syl 17 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ⊆ 𝐽)
20 simpll 789 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) → 𝐷 ∈ (∞Met‘𝑋))
21 simprl 793 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) → 𝑢𝐽)
22 simprr 795 . . . . . 6 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) → 𝑧𝑢)
231mopni2 22208 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑢𝐽𝑧𝑢) → ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
2420, 21, 22, 23syl3anc 1323 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) → ∃𝑟 ∈ ℝ+ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
25 simprl 793 . . . . . . . 8 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ+)
2625rphalfcld 11828 . . . . . . 7 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) → (𝑟 / 2) ∈ ℝ+)
27 elrp 11778 . . . . . . . 8 ((𝑟 / 2) ∈ ℝ+ ↔ ((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)))
28 nnrecl 11234 . . . . . . . 8 (((𝑟 / 2) ∈ ℝ ∧ 0 < (𝑟 / 2)) → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
2927, 28sylbi 207 . . . . . . 7 ((𝑟 / 2) ∈ ℝ+ → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
3026, 29syl 17 . . . . . 6 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) → ∃𝑛 ∈ ℕ (1 / 𝑛) < (𝑟 / 2))
313ad2antrr 761 . . . . . . . . . 10 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝐽 ∈ Top)
32 simpr1 1065 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴𝑋)
3332ad2antrr 761 . . . . . . . . . . 11 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝐴𝑋)
341mopnuni 22156 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
3534ad3antrrr 765 . . . . . . . . . . 11 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑋 = 𝐽)
3633, 35sseqtrd 3620 . . . . . . . . . 10 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝐴 𝐽)
37 simplrr 800 . . . . . . . . . . . . 13 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧𝑢)
38 simplrl 799 . . . . . . . . . . . . 13 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑢𝐽)
39 elunii 4407 . . . . . . . . . . . . 13 ((𝑧𝑢𝑢𝐽) → 𝑧 𝐽)
4037, 38, 39syl2anc 692 . . . . . . . . . . . 12 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧 𝐽)
4140, 35eleqtrrd 2701 . . . . . . . . . . 11 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧𝑋)
42 simpr3 1067 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ((cls‘𝐽)‘𝐴) = 𝑋)
4342ad2antrr 761 . . . . . . . . . . 11 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ((cls‘𝐽)‘𝐴) = 𝑋)
4441, 43eleqtrrd 2701 . . . . . . . . . 10 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧 ∈ ((cls‘𝐽)‘𝐴))
4520adantr 481 . . . . . . . . . . 11 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝐷 ∈ (∞Met‘𝑋))
46 simprrl 803 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑛 ∈ ℕ)
4746nnrpd 11814 . . . . . . . . . . . . 13 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑛 ∈ ℝ+)
4847rpreccld 11826 . . . . . . . . . . . 12 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (1 / 𝑛) ∈ ℝ+)
4948rpxrd 11817 . . . . . . . . . . 11 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (1 / 𝑛) ∈ ℝ*)
501blopn 22215 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧𝑋 ∧ (1 / 𝑛) ∈ ℝ*) → (𝑧(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
5145, 41, 49, 50syl3anc 1323 . . . . . . . . . 10 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (𝑧(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽)
52 blcntr 22128 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧𝑋 ∧ (1 / 𝑛) ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))
5345, 41, 48, 52syl3anc 1323 . . . . . . . . . 10 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑧 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))
54 eqid 2621 . . . . . . . . . . 11 𝐽 = 𝐽
5554clsndisj 20789 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝐴 𝐽𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽𝑧 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)))) → ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ≠ ∅)
5631, 36, 44, 51, 53, 55syl32anc 1331 . . . . . . . . 9 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ≠ ∅)
57 n0 3907 . . . . . . . . 9 (((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ≠ ∅ ↔ ∃𝑡 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴))
5856, 57sylib 208 . . . . . . . 8 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ∃𝑡 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴))
5946adantr 481 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑛 ∈ ℕ)
60 inss2 3812 . . . . . . . . . . . 12 ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ⊆ 𝐴
61 simpr 477 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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))
6260, 61sseldi 3581 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑡𝐴)
63 eqidd 2622 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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 6612 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → (1 / 𝑥) = (1 / 𝑛))
6564oveq2d 6620 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (𝑦(ball‘𝐷)(1 / 𝑥)) = (𝑦(ball‘𝐷)(1 / 𝑛)))
6665eqeq2d 2631 . . . . . . . . . . . 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 6611 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → (𝑦(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛)))
6867eqeq2d 2631 . . . . . . . . . . . 12 (𝑦 = 𝑡 → ((𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑛)) ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛))))
6966, 68rspc2ev 3308 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ 𝑡𝐴 ∧ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑡(ball‘𝐷)(1 / 𝑛))) → ∃𝑥 ∈ ℕ ∃𝑦𝐴 (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥)))
7059, 62, 63, 69syl3anc 1323 . . . . . . . . . 10 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → ∃𝑥 ∈ ℕ ∃𝑦𝐴 (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥)))
71 ovex 6632 . . . . . . . . . . 11 (𝑡(ball‘𝐷)(1 / 𝑛)) ∈ V
72 eqeq1 2625 . . . . . . . . . . . 12 (𝑧 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑧 = (𝑦(ball‘𝐷)(1 / 𝑥)) ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥))))
73722rexbidv 3050 . . . . . . . . . . 11 (𝑧 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (∃𝑥 ∈ ℕ ∃𝑦𝐴 𝑧 = (𝑦(ball‘𝐷)(1 / 𝑥)) ↔ ∃𝑥 ∈ ℕ ∃𝑦𝐴 (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥))))
7415rnmpt2 6723 . . . . . . . . . . 11 ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) = {𝑧 ∣ ∃𝑥 ∈ ℕ ∃𝑦𝐴 𝑧 = (𝑦(ball‘𝐷)(1 / 𝑥))}
7571, 73, 74elab2 3337 . . . . . . . . . 10 ((𝑡(ball‘𝐷)(1 / 𝑛)) ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ↔ ∃𝑥 ∈ ℕ ∃𝑦𝐴 (𝑡(ball‘𝐷)(1 / 𝑛)) = (𝑦(ball‘𝐷)(1 / 𝑥)))
7670, 75sylibr 224 . . . . . . . . 9 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))))
77 inss1 3811 . . . . . . . . . . 11 ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴) ⊆ (𝑧(ball‘𝐷)(1 / 𝑛))
7877, 61sseldi 3581 . . . . . . . . . 10 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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)))
7945adantr 481 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
8049adantr 481 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (1 / 𝑛) ∈ ℝ*)
8141adantr 481 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑧𝑋)
8233adantr 481 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝐴𝑋)
8382, 62sseldd 3584 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑡𝑋)
84 blcom 22109 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ (1 / 𝑛) ∈ ℝ*) ∧ (𝑧𝑋𝑡𝑋)) → (𝑡 ∈ (𝑧(ball‘𝐷)(1 / 𝑛)) ↔ 𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛))))
8579, 80, 81, 83, 84syl22anc 1324 . . . . . . . . . 10 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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))))
8678, 85mpbid 222 . . . . . . . . 9 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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 801 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → 𝑟 ∈ ℝ+)
8887adantr 481 . . . . . . . . . . . . 13 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑟 ∈ ℝ+)
8988rphalfcld 11828 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → (𝑟 / 2) ∈ ℝ+)
9089rpxrd 11817 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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 804 . . . . . . . . . . . . 13 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (1 / 𝑛) < (𝑟 / 2))
9287rphalfcld 11828 . . . . . . . . . . . . . 14 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (𝑟 / 2) ∈ ℝ+)
93 rpre 11783 . . . . . . . . . . . . . . 15 ((1 / 𝑛) ∈ ℝ+ → (1 / 𝑛) ∈ ℝ)
94 rpre 11783 . . . . . . . . . . . . . . 15 ((𝑟 / 2) ∈ ℝ+ → (𝑟 / 2) ∈ ℝ)
95 ltle 10070 . . . . . . . . . . . . . . 15 (((1 / 𝑛) ∈ ℝ ∧ (𝑟 / 2) ∈ ℝ) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
9693, 94, 95syl2an 494 . . . . . . . . . . . . . 14 (((1 / 𝑛) ∈ ℝ+ ∧ (𝑟 / 2) ∈ ℝ+) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
9748, 92, 96syl2anc 692 . . . . . . . . . . . . 13 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ((1 / 𝑛) < (𝑟 / 2) → (1 / 𝑛) ≤ (𝑟 / 2)))
9891, 97mpd 15 . . . . . . . . . . . 12 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (1 / 𝑛) ≤ (𝑟 / 2))
9998adantr 481 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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 22138 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡𝑋) ∧ ((1 / 𝑛) ∈ ℝ* ∧ (𝑟 / 2) ∈ ℝ*) ∧ (1 / 𝑛) ≤ (𝑟 / 2)) → (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ (𝑡(ball‘𝐷)(𝑟 / 2)))
10179, 83, 80, 90, 99, 100syl221anc 1334 . . . . . . . . . 10 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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)))
10288rpred 11816 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → 𝑟 ∈ ℝ)
103101, 86sseldd 3584 . . . . . . . . . . . 12 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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 22120 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡𝑋) ∧ (𝑟 ∈ ℝ ∧ 𝑧 ∈ (𝑡(ball‘𝐷)(𝑟 / 2)))) → (𝑡(ball‘𝐷)(𝑟 / 2)) ⊆ (𝑧(ball‘𝐷)𝑟))
10579, 83, 102, 103, 104syl22anc 1324 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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 802 . . . . . . . . . . . 12 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)
107106adantr 481 . . . . . . . . . . 11 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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)
108105, 107sstrd 3593 . . . . . . . . . 10 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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)
109101, 108sstrd 3593 . . . . . . . . 9 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((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 2687 . . . . . . . . . . 11 (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑧𝑤𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛))))
111 sseq1 3605 . . . . . . . . . . 11 (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → (𝑤𝑢 ↔ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢))
112110, 111anbi12d 746 . . . . . . . . . 10 (𝑤 = (𝑡(ball‘𝐷)(1 / 𝑛)) → ((𝑧𝑤𝑤𝑢) ↔ (𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛)) ∧ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢)))
113112rspcev 3295 . . . . . . . . 9 (((𝑡(ball‘𝐷)(1 / 𝑛)) ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∧ (𝑧 ∈ (𝑡(ball‘𝐷)(1 / 𝑛)) ∧ (𝑡(ball‘𝐷)(1 / 𝑛)) ⊆ 𝑢)) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧𝑤𝑤𝑢))
11476, 86, 109, 113syl12anc 1321 . . . . . . . 8 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) ∧ 𝑡 ∈ ((𝑧(ball‘𝐷)(1 / 𝑛)) ∩ 𝐴)) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧𝑤𝑤𝑢))
11558, 114exlimddv 1860 . . . . . . 7 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ ((𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2)))) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧𝑤𝑤𝑢))
116115anassrs 679 . . . . . 6 (((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (𝑟 / 2))) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧𝑤𝑤𝑢))
11730, 116rexlimddv 3028 . . . . 5 ((((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑧(ball‘𝐷)𝑟) ⊆ 𝑢)) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧𝑤𝑤𝑢))
11824, 117rexlimddv 3028 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) ∧ (𝑢𝐽𝑧𝑢)) → ∃𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧𝑤𝑤𝑢))
119118ralrimivva 2965 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ∀𝑢𝐽𝑧𝑢𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧𝑤𝑤𝑢))
120 basgen2 20704 . . 3 ((𝐽 ∈ Top ∧ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ⊆ 𝐽 ∧ ∀𝑢𝐽𝑧𝑢𝑤 ∈ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))(𝑧𝑤𝑤𝑢)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) = 𝐽)
1213, 19, 119, 120syl3anc 1323 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) = 𝐽)
122121, 3eqeltrd 2698 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ Top)
123 tgclb 20685 . . . 4 (ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∈ TopBases ↔ (topGen‘ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ Top)
124122, 123sylibr 224 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∈ TopBases)
125 omelon 8487 . . . . . 6 ω ∈ On
126 simpr2 1066 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐴 ≼ ω)
127 nnex 10970 . . . . . . . . 9 ℕ ∈ V
128127xpdom2 7999 . . . . . . . 8 (𝐴 ≼ ω → (ℕ × 𝐴) ≼ (ℕ × ω))
129126, 128syl 17 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ≼ (ℕ × ω))
130 nnenom 12719 . . . . . . . . 9 ℕ ≈ ω
131 omex 8484 . . . . . . . . . 10 ω ∈ V
132131enref 7932 . . . . . . . . 9 ω ≈ ω
133 xpen 8067 . . . . . . . . 9 ((ℕ ≈ ω ∧ ω ≈ ω) → (ℕ × ω) ≈ (ω × ω))
134130, 132, 133mp2an 707 . . . . . . . 8 (ℕ × ω) ≈ (ω × ω)
135 xpomen 8782 . . . . . . . 8 (ω × ω) ≈ ω
136134, 135entri 7954 . . . . . . 7 (ℕ × ω) ≈ ω
137 domentr 7959 . . . . . . 7 (((ℕ × 𝐴) ≼ (ℕ × ω) ∧ (ℕ × ω) ≈ ω) → (ℕ × 𝐴) ≼ ω)
138129, 136, 137sylancl 693 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ≼ ω)
139 ondomen 8804 . . . . . 6 ((ω ∈ On ∧ (ℕ × 𝐴) ≼ ω) → (ℕ × 𝐴) ∈ dom card)
140125, 138, 139sylancr 694 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (ℕ × 𝐴) ∈ dom card)
141 ffn 6002 . . . . . . 7 ((𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)⟶𝐽 → (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) Fn (ℕ × 𝐴))
14217, 141syl 17 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) Fn (ℕ × 𝐴))
143 dffn4 6078 . . . . . 6 ((𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) Fn (ℕ × 𝐴) ↔ (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)–onto→ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))))
144142, 143sylib 208 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)–onto→ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))))
145 fodomnum 8824 . . . . 5 ((ℕ × 𝐴) ∈ dom card → ((𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))):(ℕ × 𝐴)–onto→ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) → ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ (ℕ × 𝐴)))
146140, 144, 145sylc 65 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ (ℕ × 𝐴))
147 domtr 7953 . . . 4 ((ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ (ℕ × 𝐴) ∧ (ℕ × 𝐴) ≼ ω) → ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ ω)
148146, 138, 147syl2anc 692 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ ω)
149 2ndci 21161 . . 3 ((ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ∈ TopBases ∧ ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥))) ≼ ω) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ 2nd𝜔)
150124, 148, 149syl2anc 692 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → (topGen‘ran (𝑥 ∈ ℕ, 𝑦𝐴 ↦ (𝑦(ball‘𝐷)(1 / 𝑥)))) ∈ 2nd𝜔)
151121, 150eqeltrrd 2699 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ 2nd𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  cin 3554  wss 3555  c0 3891   cuni 4402   class class class wbr 4613   × cxp 5072  dom cdm 5074  ran crn 5075  Oncon0 5682   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  (class class class)co 6604  cmpt2 6606  ωcom 7012  cen 7896  cdom 7897  cardccrd 8705  cr 9879  0cc0 9880  1c1 9881  *cxr 10017   < clt 10018  cle 10019   / cdiv 10628  cn 10964  2c2 11014  +crp 11776  topGenctg 16019  ∞Metcxmt 19650  ballcbl 19652  MetOpencmopn 19655  Topctop 20617  TopBasesctb 20620  clsccl 20732  2nd𝜔c2ndc 21151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-acn 8712  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-ntr 20734  df-cls 20735  df-2ndc 21153
This theorem is referenced by:  met2ndc  22238
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