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Theorem heibor 32790
Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 32779 and heiborlem1 32780 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.)
Hypothesis
Ref Expression
heibor.1 𝐽 = (MetOpen‘𝐷)
Assertion
Ref Expression
heibor ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Proof of Theorem heibor
Dummy variables 𝑡 𝑛 𝑦 𝑘 𝑟 𝑢 𝑚 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 heibor.1 . . 3 𝐽 = (MetOpen‘𝐷)
21heibor1 32779 . 2 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
3 cmetmet 22892 . . . 4 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
43adantr 480 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐷 ∈ (Met‘𝑋))
5 metxmet 21949 . . . . . 6 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
61mopntop 22055 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
73, 5, 63syl 18 . . . . 5 (𝐷 ∈ (CMet‘𝑋) → 𝐽 ∈ Top)
87adantr 480 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Top)
9 istotbnd 32738 . . . . . . . . . . . . 13 (𝐷 ∈ (TotBnd‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑟 ∈ ℝ+𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟))))
109simprbi 479 . . . . . . . . . . . 12 (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ ℝ+𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)))
11 2nn 11062 . . . . . . . . . . . . . . 15 2 ∈ ℕ
12 nnexpcl 12735 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1311, 12mpan 702 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℕ)
1413nnrpd 11746 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℝ+)
1514rpreccld 11758 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (1 / (2↑𝑛)) ∈ ℝ+)
16 oveq2 6557 . . . . . . . . . . . . . . . . . 18 (𝑟 = (1 / (2↑𝑛)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
1716eqeq2d 2620 . . . . . . . . . . . . . . . . 17 (𝑟 = (1 / (2↑𝑛)) → (𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
1817rexbidv 3034 . . . . . . . . . . . . . . . 16 (𝑟 = (1 / (2↑𝑛)) → (∃𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
1918ralbidv 2969 . . . . . . . . . . . . . . 15 (𝑟 = (1 / (2↑𝑛)) → (∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
2019anbi2d 736 . . . . . . . . . . . . . 14 (𝑟 = (1 / (2↑𝑛)) → (( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)) ↔ ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
2120rexbidv 3034 . . . . . . . . . . . . 13 (𝑟 = (1 / (2↑𝑛)) → (∃𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)) ↔ ∃𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
2221rspccva 3281 . . . . . . . . . . . 12 ((∀𝑟 ∈ ℝ+𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)) ∧ (1 / (2↑𝑛)) ∈ ℝ+) → ∃𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
2310, 15, 22syl2an 493 . . . . . . . . . . 11 ((𝐷 ∈ (TotBnd‘𝑋) ∧ 𝑛 ∈ ℕ0) → ∃𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
2423expcom 450 . . . . . . . . . 10 (𝑛 ∈ ℕ0 → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
2524adantl 481 . . . . . . . . 9 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
26 oveq1 6556 . . . . . . . . . . . . . . 15 (𝑦 = (𝑚𝑣) → (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
2726eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑦 = (𝑚𝑣) → (𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
2827ac6sfi 8089 . . . . . . . . . . . . 13 ((𝑢 ∈ Fin ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑚(𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
2928adantrl 748 . . . . . . . . . . . 12 ((𝑢 ∈ Fin ∧ ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → ∃𝑚(𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
3029adantl 481 . . . . . . . . . . 11 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))) → ∃𝑚(𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
31 simp3l 1082 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑚:𝑢𝑋)
32 frn 5966 . . . . . . . . . . . . . . . . . . 19 (𝑚:𝑢𝑋 → ran 𝑚𝑋)
3331, 32syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚𝑋)
341mopnuni 22056 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
353, 5, 343syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (CMet‘𝑋) → 𝑋 = 𝐽)
3635adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋 = 𝐽)
37363ad2ant1 1075 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑋 = 𝐽)
3833, 37sseqtrd 3604 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 𝐽)
39 fvex 6113 . . . . . . . . . . . . . . . . . . . 20 (MetOpen‘𝐷) ∈ V
401, 39eqeltri 2684 . . . . . . . . . . . . . . . . . . 19 𝐽 ∈ V
4140uniex 6851 . . . . . . . . . . . . . . . . . 18 𝐽 ∈ V
4241elpw2 4755 . . . . . . . . . . . . . . . . 17 (ran 𝑚 ∈ 𝒫 𝐽 ↔ ran 𝑚 𝐽)
4338, 42sylibr 223 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ∈ 𝒫 𝐽)
44 simp2l 1080 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑢 ∈ Fin)
45 ffn 5958 . . . . . . . . . . . . . . . . . . 19 (𝑚:𝑢𝑋𝑚 Fn 𝑢)
46 dffn4 6034 . . . . . . . . . . . . . . . . . . 19 (𝑚 Fn 𝑢𝑚:𝑢onto→ran 𝑚)
4745, 46sylib 207 . . . . . . . . . . . . . . . . . 18 (𝑚:𝑢𝑋𝑚:𝑢onto→ran 𝑚)
48 fofi 8135 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ Fin ∧ 𝑚:𝑢onto→ran 𝑚) → ran 𝑚 ∈ Fin)
4947, 48sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ Fin ∧ 𝑚:𝑢𝑋) → ran 𝑚 ∈ Fin)
5044, 31, 49syl2anc 691 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ∈ Fin)
5143, 50elind 3760 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ∈ (𝒫 𝐽 ∩ Fin))
5226eleq2d 2673 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑚𝑣) → (𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
5352rexrn 6269 . . . . . . . . . . . . . . . . . . 19 (𝑚 Fn 𝑢 → (∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑣𝑢 𝑟 ∈ ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
54 eliun 4460 . . . . . . . . . . . . . . . . . . 19 (𝑟 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
55 eliun 4460 . . . . . . . . . . . . . . . . . . 19 (𝑟 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑣𝑢 𝑟 ∈ ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
5653, 54, 553bitr4g 302 . . . . . . . . . . . . . . . . . 18 (𝑚 Fn 𝑢 → (𝑟 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
5756eqrdv 2608 . . . . . . . . . . . . . . . . 17 (𝑚 Fn 𝑢 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
5831, 45, 573syl 18 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
59 simp3r 1083 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
60 uniiun 4509 . . . . . . . . . . . . . . . . . 18 𝑢 = 𝑣𝑢 𝑣
61 iuneq2 4473 . . . . . . . . . . . . . . . . . 18 (∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → 𝑣𝑢 𝑣 = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
6260, 61syl5eq 2656 . . . . . . . . . . . . . . . . 17 (∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → 𝑢 = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
6359, 62syl 17 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑢 = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
64 simp2r 1081 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑢 = 𝑋)
6558, 63, 643eqtr2rd 2651 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑋 = 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
66 iuneq1 4470 . . . . . . . . . . . . . . . . 17 (𝑡 = ran 𝑚 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
6766eqeq2d 2620 . . . . . . . . . . . . . . . 16 (𝑡 = ran 𝑚 → (𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑋 = 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
6867rspcev 3282 . . . . . . . . . . . . . . 15 ((ran 𝑚 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑋 = 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
6951, 65, 68syl2anc 691 . . . . . . . . . . . . . 14 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ∃𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
70693expia 1259 . . . . . . . . . . . . 13 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋)) → ((𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7170adantrrr 757 . . . . . . . . . . . 12 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))) → ((𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7271exlimdv 1848 . . . . . . . . . . 11 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))) → (∃𝑚(𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7330, 72mpd 15 . . . . . . . . . 10 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))) → ∃𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
7473rexlimdvaa 3014 . . . . . . . . 9 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → (∃𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7525, 74syld 46 . . . . . . . 8 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7675ralrimdva 2952 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑛 ∈ ℕ0𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7741pwex 4774 . . . . . . . . 9 𝒫 𝐽 ∈ V
7877inex1 4727 . . . . . . . 8 (𝒫 𝐽 ∩ Fin) ∈ V
79 nn0ennn 12640 . . . . . . . . 9 0 ≈ ℕ
80 nnenom 12641 . . . . . . . . 9 ℕ ≈ ω
8179, 80entri 7896 . . . . . . . 8 0 ≈ ω
82 iuneq1 4470 . . . . . . . . 9 (𝑡 = (𝑚𝑛) → 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
8382eqeq2d 2620 . . . . . . . 8 (𝑡 = (𝑚𝑛) → (𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
8478, 81, 83axcc4 9144 . . . . . . 7 (∀𝑛 ∈ ℕ0𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) → ∃𝑚(𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
8576, 84syl6 34 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑚(𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
86 elpwi 4117 . . . . . . . . . 10 (𝑟 ∈ 𝒫 𝐽𝑟𝐽)
87 eqid 2610 . . . . . . . . . . . 12 {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣} = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣}
88 eqid 2610 . . . . . . . . . . . 12 {⟨𝑡, 𝑘⟩ ∣ (𝑘 ∈ ℕ0𝑡 ∈ (𝑚𝑘) ∧ (𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘) ∈ {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣})} = {⟨𝑡, 𝑘⟩ ∣ (𝑘 ∈ ℕ0𝑡 ∈ (𝑚𝑘) ∧ (𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘) ∈ {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣})}
89 eqid 2610 . . . . . . . . . . . 12 (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
90 simpl 472 . . . . . . . . . . . 12 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → 𝐷 ∈ (CMet‘𝑋))
9135pweqd 4113 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (CMet‘𝑋) → 𝒫 𝑋 = 𝒫 𝐽)
9291ineq1d 3775 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝑋 ∩ Fin) = (𝒫 𝐽 ∩ Fin))
9392feq3d 5945 . . . . . . . . . . . . . 14 (𝐷 ∈ (CMet‘𝑋) → (𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin) ↔ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)))
9493biimpar 501 . . . . . . . . . . . . 13 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin))
9594adantrr 749 . . . . . . . . . . . 12 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin))
96 oveq1 6556 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑦 → (𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑦(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛))
9796cbviunv 4495 . . . . . . . . . . . . . . . . . 18 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = 𝑦 ∈ (𝑚𝑛)(𝑦(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛)
98 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) → 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin))
99 inss1 3795 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝐽
10099, 91syl5sseqr 3617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝑋)
101 fss 5969 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝑋) → 𝑚:ℕ0⟶𝒫 𝑋)
10298, 100, 101syl2anr 494 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶𝒫 𝑋)
103102ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚𝑛) ∈ 𝒫 𝑋)
104103elpwid 4118 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚𝑛) ⊆ 𝑋)
105104sselda 3568 . . . . . . . . . . . . . . . . . . . 20 ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚𝑛)) → 𝑦𝑋)
106 simplr 788 . . . . . . . . . . . . . . . . . . . 20 ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚𝑛)) → 𝑛 ∈ ℕ0)
107 oveq1 6556 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑦 → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑚))))
108 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
109108oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (1 / (2↑𝑚)) = (1 / (2↑𝑛)))
110109oveq2d 6565 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (𝑦(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
111 ovex 6577 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ∈ V
112107, 110, 89, 111ovmpt2 6694 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑋𝑛 ∈ ℕ0) → (𝑦(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
113105, 106, 112syl2anc 691 . . . . . . . . . . . . . . . . . . 19 ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚𝑛)) → (𝑦(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
114113iuneq2dv 4478 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → 𝑦 ∈ (𝑚𝑛)(𝑦(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
11597, 114syl5eq 2656 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
116115eqeq2d 2620 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑋 = 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) ↔ 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
117116biimprd 237 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))) → 𝑋 = 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛)))
118117ralimdva 2945 . . . . . . . . . . . . . 14 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) → (∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))) → ∀𝑛 ∈ ℕ0 𝑋 = 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛)))
119118impr 647 . . . . . . . . . . . . 13 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → ∀𝑛 ∈ ℕ0 𝑋 = 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛))
120 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑚𝑛) = (𝑚𝑘))
121120iuneq1d 4481 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = 𝑡 ∈ (𝑚𝑘)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛))
122 simpl 472 . . . . . . . . . . . . . . . . . 18 ((𝑛 = 𝑘𝑡 ∈ (𝑚𝑘)) → 𝑛 = 𝑘)
123122oveq2d 6565 . . . . . . . . . . . . . . . . 17 ((𝑛 = 𝑘𝑡 ∈ (𝑚𝑘)) → (𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
124123iuneq2dv 4478 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 𝑡 ∈ (𝑚𝑘)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = 𝑡 ∈ (𝑚𝑘)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
125121, 124eqtrd 2644 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = 𝑡 ∈ (𝑚𝑘)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
126125eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝑋 = 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) ↔ 𝑋 = 𝑡 ∈ (𝑚𝑘)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘)))
127126cbvralv 3147 . . . . . . . . . . . . 13 (∀𝑛 ∈ ℕ0 𝑋 = 𝑡 ∈ (𝑚𝑛)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) ↔ ∀𝑘 ∈ ℕ0 𝑋 = 𝑡 ∈ (𝑚𝑘)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
128119, 127sylib 207 . . . . . . . . . . . 12 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → ∀𝑘 ∈ ℕ0 𝑋 = 𝑡 ∈ (𝑚𝑘)(𝑡(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
1291, 87, 88, 89, 90, 95, 128heiborlem10 32789 . . . . . . . . . . 11 (((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) ∧ (𝑟𝐽 𝐽 = 𝑟)) → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)
130129exp32 629 . . . . . . . . . 10 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → (𝑟𝐽 → ( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)))
13186, 130syl5 33 . . . . . . . . 9 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → (𝑟 ∈ 𝒫 𝐽 → ( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)))
132131ralrimiv 2948 . . . . . . . 8 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣))
133132ex 449 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → ((𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)))
134133exlimdv 1848 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → (∃𝑚(𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)))
13585, 134syld 46 . . . . 5 (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)))
136135imp 444 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣))
137 eqid 2610 . . . . 5 𝐽 = 𝐽
138137iscmp 21001 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)))
1398, 136, 138sylanbrc 695 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Comp)
1404, 139jca 553 . 2 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → (𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp))
1412, 140impbii 198 1 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wral 2896  wrex 2897  Vcvv 3173  cin 3539  wss 3540  𝒫 cpw 4108   cuni 4372   ciun 4455  {copab 4642  ran crn 5039   Fn wfn 5799  wf 5800  ontowfo 5802  cfv 5804  (class class class)co 6549  cmpt2 6551  ωcom 6957  Fincfn 7841  1c1 9816   / cdiv 10563  cn 10897  2c2 10947  0cn0 11169  +crp 11708  cexp 12722  ∞Metcxmt 19552  Metcme 19553  ballcbl 19554  MetOpencmopn 19557  Topctop 20517  Compccmp 20999  CMetcms 22860  TotBndctotbnd 32735
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-cc 9140  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-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  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-3 10957  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-ico 12052  df-icc 12053  df-fz 12198  df-fl 12455  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lm 20843  df-haus 20929  df-cmp 21000  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-cfil 22861  df-cau 22862  df-cmet 22863  df-totbnd 32737
This theorem is referenced by:  rrnheibor  32806
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