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Theorem heibor 37283
Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 37272 and heiborlem1 37273 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 37272 . 2 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
3 cmetmet 25207 . . . 4 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
43adantr 480 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐷 ∈ (Met‘𝑋))
5 metxmet 24233 . . . . . 6 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
61mopntop 24339 . . . . . 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 37231 . . . . . . . . . . . . 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 12309 . . . . . . . . . . . . . . 15 2 ∈ ℕ
12 nnexpcl 14065 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1311, 12mpan 689 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℕ)
1413nnrpd 13040 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℝ+)
1514rpreccld 13052 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (1 / (2↑𝑛)) ∈ ℝ+)
16 oveq2 7422 . . . . . . . . . . . . . . . . . 18 (𝑟 = (1 / (2↑𝑛)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
1716eqeq2d 2738 . . . . . . . . . . . . . . . . 17 (𝑟 = (1 / (2↑𝑛)) → (𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
1817rexbidv 3173 . . . . . . . . . . . . . . . 16 (𝑟 = (1 / (2↑𝑛)) → (∃𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
1918ralbidv 3172 . . . . . . . . . . . . . . 15 (𝑟 = (1 / (2↑𝑛)) → (∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
2019anbi2d 628 . . . . . . . . . . . . . 14 (𝑟 = (1 / (2↑𝑛)) → (( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)) ↔ ( 𝑢 = 𝑋 ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
2120rexbidv 3173 . . . . . . . . . . . . 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 3606 . . . . . . . . . . . 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 595 . . . . . . . . . . 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 7421 . . . . . . . . . . . . . . 15 (𝑦 = (𝑚𝑣) → (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
2726eqeq2d 2738 . . . . . . . . . . . . . 14 (𝑦 = (𝑚𝑣) → (𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
2827ac6sfi 9305 . . . . . . . . . . . . 13 ((𝑢 ∈ Fin ∧ ∀𝑣𝑢𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑚(𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
2928adantrl 715 . . . . . . . . . . . 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 1199 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑚:𝑢𝑋)
3231frnd 6724 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚𝑋)
331mopnuni 24340 . . . . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑋 = 𝐽)
3732, 36sseqtrd 4018 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 𝐽)
381fvexi 6905 . . . . . . . . . . . . . . . . . . 19 𝐽 ∈ V
3938uniex 7740 . . . . . . . . . . . . . . . . . 18 𝐽 ∈ V
4039elpw2 5341 . . . . . . . . . . . . . . . . 17 (ran 𝑚 ∈ 𝒫 𝐽 ↔ ran 𝑚 𝐽)
4137, 40sylibr 233 . . . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑢 ∈ Fin)
43 ffn 6716 . . . . . . . . . . . . . . . . . . 19 (𝑚:𝑢𝑋𝑚 Fn 𝑢)
44 dffn4 6811 . . . . . . . . . . . . . . . . . . 19 (𝑚 Fn 𝑢𝑚:𝑢onto→ran 𝑚)
4543, 44sylib 217 . . . . . . . . . . . . . . . . . 18 (𝑚:𝑢𝑋𝑚:𝑢onto→ran 𝑚)
46 fofi 9356 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ Fin ∧ 𝑚:𝑢onto→ran 𝑚) → ran 𝑚 ∈ Fin)
4745, 46sylan2 592 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ Fin ∧ 𝑚:𝑢𝑋) → ran 𝑚 ∈ Fin)
4842, 31, 47syl2anc 583 . . . . . . . . . . . . . . . 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 4190 . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑚𝑣) → (𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
5150rexrn 7091 . . . . . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . . . 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 5055 . . . . . . . . . . . . . . . . . 18 𝑢 = 𝑣𝑢 𝑣
59 iuneq2 5010 . . . . . . . . . . . . . . . . . 18 (∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → 𝑣𝑢 𝑣 = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
6058, 59eqtrid 2779 . . . . . . . . . . . . . . . . 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 1198 . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . 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 5007 . . . . . . . . . . . . . . . 16 (𝑡 = ran 𝑚 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
6564rspceeqv 3629 . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . 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 724 . . . . . . . . . . . 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 1929 . . . . . . . . . . 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 3151 . . . . . . . . 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 3149 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑛 ∈ ℕ0𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7439pwex 5374 . . . . . . . . 9 𝒫 𝐽 ∈ V
7574inex1 5311 . . . . . . . 8 (𝒫 𝐽 ∩ Fin) ∈ V
76 nn0ennn 13970 . . . . . . . . 9 0 ≈ ℕ
77 nnenom 13971 . . . . . . . . 9 ℕ ≈ ω
7876, 77entri 9022 . . . . . . . 8 0 ≈ ω
79 iuneq1 5007 . . . . . . . . 9 (𝑡 = (𝑚𝑛) → 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
8079eqeq2d 2738 . . . . . . . 8 (𝑡 = (𝑚𝑛) → (𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
8175, 78, 80axcc4 10456 . . . . . . 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 4605 . . . . . . . . . 10 (𝑟 ∈ 𝒫 𝐽𝑟𝐽)
84 eqid 2727 . . . . . . . . . . . 12 {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣} = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣}
85 eqid 2727 . . . . . . . . . . . 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 2727 . . . . . . . . . . . 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 4615 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (CMet‘𝑋) → 𝒫 𝑋 = 𝒫 𝐽)
8988ineq1d 4207 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝑋 ∩ Fin) = (𝒫 𝐽 ∩ Fin))
9089feq3d 6703 . . . . . . . . . . . . . 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 716 . . . . . . . . . . . 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 7421 . . . . . . . . . . . . . . . . . . 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 5037 . . . . . . . . . . . . . . . . . 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 4224 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝐽
9796, 88sseqtrrid 4031 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝑋)
98 fss 6733 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin) ∧ (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝑋) → 𝑚:ℕ0⟶𝒫 𝑋)
9995, 97, 98syl2anr 596 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶𝒫 𝑋)
10099ffvelcdmda 7088 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚𝑛) ∈ 𝒫 𝑋)
101100elpwid 4607 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚𝑛) ⊆ 𝑋)
102101sselda 3978 . . . . . . . . . . . . . . . . . . . 20 ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚𝑛)) → 𝑦𝑋)
103 simplr 768 . . . . . . . . . . . . . . . . . . . 20 ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚𝑛)) → 𝑛 ∈ ℕ0)
104 oveq1 7421 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑦 → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑚))))
105 oveq2 7422 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
106105oveq2d 7430 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (1 / (2↑𝑚)) = (1 / (2↑𝑛)))
107106oveq2d 7430 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (𝑦(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
108 ovex 7447 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ∈ V
109104, 107, 86, 108ovmpo 7575 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑋𝑛 ∈ ℕ0) → (𝑦(𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
110102, 103, 109syl2anc 583 . . . . . . . . . . . . . . . . . . 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 5015 . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . . 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 3162 . . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑚𝑛) = (𝑚𝑘))
118117iuneq1d 5018 . . . . . . . . . . . . . . . 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 7430 . . . . . . . . . . . . . . . . 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 5015 . . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . 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 3229 . . . . . . . . . . . . 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 217 . . . . . . . . . . . 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 37282 . . . . . . . . . . 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 3140 . . . . . . . 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 1929 . . . . . 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 2727 . . . . 5 𝐽 = 𝐽
135134iscmp 23285 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽( 𝐽 = 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin) 𝐽 = 𝑣)))
1368, 133, 135sylanbrc 582 . . 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 208 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 205  wa 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  {cab 2704  wral 3056  wrex 3065  cin 3943  wss 3944  𝒫 cpw 4598   cuni 4903   ciun 4991  {copab 5204  ran crn 5673   Fn wfn 6537  wf 6538  ontowfo 6540  cfv 6542  (class class class)co 7414  cmpo 7416  ωcom 7864  Fincfn 8957  1c1 11133   / cdiv 11895  cn 12236  2c2 12291  0cn0 12496  +crp 13000  cexp 14052  ∞Metcxmet 21257  Metcmet 21258  ballcbl 21259  MetOpencmopn 21262  Topctop 22788  Compccmp 23283  CMetccmet 25175  TotBndctotbnd 37228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-cc 10452  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8718  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fi 9428  df-sup 9459  df-inf 9460  df-oi 9527  df-card 9956  df-acn 9959  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-z 12583  df-uz 12847  df-q 12957  df-rp 13001  df-xneg 13118  df-xadd 13119  df-xmul 13120  df-ico 13356  df-icc 13357  df-fz 13511  df-fl 13783  df-seq 13993  df-exp 14053  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15458  df-rlim 15459  df-rest 17397  df-topgen 17418  df-psmet 21264  df-xmet 21265  df-met 21266  df-bl 21267  df-mopn 21268  df-fbas 21269  df-fg 21270  df-top 22789  df-topon 22806  df-bases 22842  df-cld 22916  df-ntr 22917  df-cls 22918  df-nei 22995  df-lm 23126  df-haus 23212  df-cmp 23284  df-fil 23743  df-fm 23835  df-flim 23836  df-flf 23837  df-cfil 25176  df-cau 25177  df-cmet 25178  df-totbnd 37230
This theorem is referenced by:  rrnheibor  37299
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