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Theorem heibor 37828
Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 37817 and heiborlem1 37818 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 37817 . 2 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
3 cmetmet 25320 . . . 4 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
43adantr 480 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐷 ∈ (Met‘𝑋))
5 metxmet 24344 . . . . . 6 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
61mopntop 24450 . . . . . 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 37776 . . . . . . . . . . . . 13 (𝐷 ∈ (TotBnd‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑟 ∈ ℝ+𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟))))
109simprbi 496 . . . . . . . . . . . 12 (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ ℝ+𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)))
11 2nn 12339 . . . . . . . . . . . . . . 15 2 ∈ ℕ
12 nnexpcl 14115 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1311, 12mpan 690 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℕ)
1413nnrpd 13075 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℝ+)
1514rpreccld 13087 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (1 / (2↑𝑛)) ∈ ℝ+)
16 oveq2 7439 . . . . . . . . . . . . . . . . . 18 (𝑟 = (1 / (2↑𝑛)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
1716eqeq2d 2748 . . . . . . . . . . . . . . . . 17 (𝑟 = (1 / (2↑𝑛)) → (𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
1817rexbidv 3179 . . . . . . . . . . . . . . . 16 (𝑟 = (1 / (2↑𝑛)) → (∃𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
1918ralbidv 3178 . . . . . . . . . . . . . . 15 (𝑟 = (1 / (2↑𝑛)) → (∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
2019anbi2d 630 . . . . . . . . . . . . . 14 (𝑟 = (1 / (2↑𝑛)) → (( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)) ↔ ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
2120rexbidv 3179 . . . . . . . . . . . . 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 3621 . . . . . . . . . . . 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 596 . . . . . . . . . . 11 ((𝐷 ∈ (TotBnd‘𝑋) ∧ 𝑛 ∈ ℕ0) → ∃𝑢 ∈ Fin ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
2423expcom 413 . . . . . . . . . 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 7438 . . . . . . . . . . . . . . 15 (𝑦 = (𝑚𝑣) → (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
2726eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑦 = (𝑚𝑣) → (𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
2827ac6sfi 9320 . . . . . . . . . . . . 13 ((𝑢 ∈ Fin ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑚(𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
2928adantrl 716 . . . . . . . . . . . 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 1202 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑚:𝑢𝑋)
3231frnd 6744 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚𝑋)
331mopnuni 24451 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
343, 5, 333syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (CMet‘𝑋) → 𝑋 = 𝐽)
3534adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋 = 𝐽)
36353ad2ant1 1134 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑋 = 𝐽)
3732, 36sseqtrd 4020 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 𝐽)
381fvexi 6920 . . . . . . . . . . . . . . . . . . 19 𝐽 ∈ V
3938uniex 7761 . . . . . . . . . . . . . . . . . 18 𝐽 ∈ V
4039elpw2 5334 . . . . . . . . . . . . . . . . 17 (ran 𝑚 ∈ 𝒫 𝐽 ↔ ran 𝑚 𝐽)
4137, 40sylibr 234 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ∈ 𝒫 𝐽)
42 simp2l 1200 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑢 ∈ Fin)
43 ffn 6736 . . . . . . . . . . . . . . . . . . 19 (𝑚:𝑢𝑋𝑚 Fn 𝑢)
44 dffn4 6826 . . . . . . . . . . . . . . . . . . 19 (𝑚 Fn 𝑢𝑚:𝑢onto→ran 𝑚)
4543, 44sylib 218 . . . . . . . . . . . . . . . . . 18 (𝑚:𝑢𝑋𝑚:𝑢onto→ran 𝑚)
46 fofi 9351 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ Fin ∧ 𝑚:𝑢onto→ran 𝑚) → ran 𝑚 ∈ Fin)
4745, 46sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ Fin ∧ 𝑚:𝑢𝑋) → ran 𝑚 ∈ Fin)
4842, 31, 47syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ∈ Fin)
4941, 48elind 4200 . . . . . . . . . . . . . . 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))
5026eleq2d 2827 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑚𝑣) → (𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
5150rexrn 7107 . . . . . . . . . . . . . . . . . . 19 (𝑚 Fn 𝑢 → (∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑣𝑢 𝑟 ∈ ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
52 eliun 4995 . . . . . . . . . . . . . . . . . . 19 (𝑟 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
53 eliun 4995 . . . . . . . . . . . . . . . . . . 19 (𝑟 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑣𝑢 𝑟 ∈ ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
5451, 52, 533bitr4g 314 . . . . . . . . . . . . . . . . . 18 (𝑚 Fn 𝑢 → (𝑟 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
5554eqrdv 2735 . . . . . . . . . . . . . . . . 17 (𝑚 Fn 𝑢 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
5631, 43, 553syl 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))))
57 simp3r 1203 . . . . . . . . . . . . . . . . 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))))
58 uniiun 5058 . . . . . . . . . . . . . . . . . 18 𝑢 = 𝑣𝑢 𝑣
59 iuneq2 5011 . . . . . . . . . . . . . . . . . 18 (∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → 𝑣𝑢 𝑣 = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
6058, 59eqtrid 2789 . . . . . . . . . . . . . . . . 17 (∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → 𝑢 = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
6157, 60syl 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))))
62 simp2r 1201 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑢 = 𝑋)
6356, 61, 623eqtr2rd 2784 . . . . . . . . . . . . . . 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))))
64 iuneq1 5008 . . . . . . . . . . . . . . . 16 (𝑡 = ran 𝑚 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
6564rspceeqv 3645 . . . . . . . . . . . . . . 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))))
6649, 63, 65syl2anc 584 . . . . . . . . . . . . . 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))))
67663expia 1122 . . . . . . . . . . . . 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)))))
6867adantrrr 725 . . . . . . . . . . . 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)))))
6968exlimdv 1933 . . . . . . . . . . 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)))))
7030, 69mpd 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))))
7170rexlimdvaa 3156 . . . . . . . . 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)))))
7225, 71syld 47 . . . . . . . 8 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7372ralrimdva 3154 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑛 ∈ ℕ0𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7439pwex 5380 . . . . . . . . 9 𝒫 𝐽 ∈ V
7574inex1 5317 . . . . . . . 8 (𝒫 𝐽 ∩ Fin) ∈ V
76 nn0ennn 14020 . . . . . . . . 9 0 ≈ ℕ
77 nnenom 14021 . . . . . . . . 9 ℕ ≈ ω
7876, 77entri 9048 . . . . . . . 8 0 ≈ ω
79 iuneq1 5008 . . . . . . . . 9 (𝑡 = (𝑚𝑛) → 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
8079eqeq2d 2748 . . . . . . . 8 (𝑡 = (𝑚𝑛) → (𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
8175, 78, 80axcc4 10479 . . . . . . 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)))))
8273, 81syl6 35 . . . . . 6 (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑚(𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
83 elpwi 4607 . . . . . . . . . 10 (𝑟 ∈ 𝒫 𝐽𝑟𝐽)
84 eqid 2737 . . . . . . . . . . . 12 {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣} = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣}
85 eqid 2737 . . . . . . . . . . . 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})}
86 eqid 2737 . . . . . . . . . . . 12 (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
87 simpl 482 . . . . . . . . . . . 12 ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → 𝐷 ∈ (CMet‘𝑋))
8834pweqd 4617 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (CMet‘𝑋) → 𝒫 𝑋 = 𝒫 𝐽)
8988ineq1d 4219 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝑋 ∩ Fin) = (𝒫 𝐽 ∩ Fin))
9089feq3d 6723 . . . . . . . . . . . . . 14 (𝐷 ∈ (CMet‘𝑋) → (𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin) ↔ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)))
9190biimpar 477 . . . . . . . . . . . . 13 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin))
9291adantrr 717 . . . . . . . . . . . 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))
93 oveq1 7438 . . . . . . . . . . . . . . . . . . 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))
9493cbviunv 5040 . . . . . . . . . . . . . . . . . 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)
95 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) → 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin))
96 inss1 4237 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝐽
9796, 88sseqtrrid 4027 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝑋)
98 fss 6752 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝑋) → 𝑚:ℕ0⟶𝒫 𝑋)
9995, 97, 98syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶𝒫 𝑋)
10099ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚𝑛) ∈ 𝒫 𝑋)
101100elpwid 4609 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚𝑛) ⊆ 𝑋)
102101sselda 3983 . . . . . . . . . . . . . . . . . . . 20 ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚𝑛)) → 𝑦𝑋)
103 simplr 769 . . . . . . . . . . . . . . . . . . . 20 ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚𝑛)) → 𝑛 ∈ ℕ0)
104 oveq1 7438 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑦 → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑚))))
105 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
106105oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (1 / (2↑𝑚)) = (1 / (2↑𝑛)))
107106oveq2d 7447 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (𝑦(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
108 ovex 7464 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ∈ V
109104, 107, 86, 108ovmpo 7593 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑋𝑛 ∈ ℕ0) → (𝑦(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
110102, 103, 109syl2anc 584 . . . . . . . . . . . . . . . . . . 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))))
111110iuneq2dv 5016 . . . . . . . . . . . . . . . . . 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))))
11294, 111eqtrid 2789 . . . . . . . . . . . . . . . . 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))))
113112eqeq2d 2748 . . . . . . . . . . . . . . . 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)))))
114113biimprd 248 . . . . . . . . . . . . . . 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)))
115114ralimdva 3167 . . . . . . . . . . . . . 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)))
116115impr 454 . . . . . . . . . . . . 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))
117 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑚𝑛) = (𝑚𝑘))
118117iuneq1d 5019 . . . . . . . . . . . . . . . 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))
119 simpl 482 . . . . . . . . . . . . . . . . . 18 ((𝑛 = 𝑘𝑡 ∈ (𝑚𝑘)) → 𝑛 = 𝑘)
120119oveq2d 7447 . . . . . . . . . . . . . . . . 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))
121120iuneq2dv 5016 . . . . . . . . . . . . . . . 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))
122118, 121eqtrd 2777 . . . . . . . . . . . . . . 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))
123122eqeq2d 2748 . . . . . . . . . . . . . 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)))
124123cbvralvw 3237 . . . . . . . . . . . . 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))
125116, 124sylib 218 . . . . . . . . . . . 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))
1261, 84, 85, 86, 87, 92, 125heiborlem10 37827 . . . . . . . . . . 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)
127126exp32 420 . . . . . . . . . 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)))
12883, 127syl5 34 . . . . . . . . 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)))
129128ralrimiv 3145 . . . . . . . 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))
130129ex 412 . . . . . . 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)))
131130exlimdv 1933 . . . . . 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)))
13282, 131syld 47 . . . . 5 (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)))
133132imp 406 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣))
134 eqid 2737 . . . . 5 𝐽 = 𝐽
135134iscmp 23396 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)))
1368, 133, 135sylanbrc 583 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Comp)
1374, 136jca 511 . 2 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → (𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp))
1382, 137impbii 209 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 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   ciun 4991  {copab 5205  ran crn 5686   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  Fincfn 8985  1c1 11156   / cdiv 11920  cn 12266  2c2 12321  0cn0 12526  +crp 13034  cexp 14102  ∞Metcxmet 21349  Metcmet 21350  ballcbl 21351  MetOpencmopn 21354  Topctop 22899  Compccmp 23394  CMetccmet 25288  TotBndctotbnd 37773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ico 13393  df-icc 13394  df-fz 13548  df-fl 13832  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lm 23237  df-haus 23323  df-cmp 23395  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-cfil 25289  df-cau 25290  df-cmet 25291  df-totbnd 37775
This theorem is referenced by:  rrnheibor  37844
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