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Theorem heibor 37807
Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 37796 and heiborlem1 37797 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 37796 . 2 ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
3 cmetmet 25333 . . . 4 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
43adantr 480 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐷 ∈ (Met‘𝑋))
5 metxmet 24359 . . . . . 6 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
61mopntop 24465 . . . . . 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 37755 . . . . . . . . . . . . 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 12336 . . . . . . . . . . . . . . 15 2 ∈ ℕ
12 nnexpcl 14111 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1311, 12mpan 690 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℕ)
1413nnrpd 13072 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℝ+)
1514rpreccld 13084 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (1 / (2↑𝑛)) ∈ ℝ+)
16 oveq2 7438 . . . . . . . . . . . . . . . . . 18 (𝑟 = (1 / (2↑𝑛)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
1716eqeq2d 2745 . . . . . . . . . . . . . . . . 17 (𝑟 = (1 / (2↑𝑛)) → (𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
1817rexbidv 3176 . . . . . . . . . . . . . . . 16 (𝑟 = (1 / (2↑𝑛)) → (∃𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
1918ralbidv 3175 . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . 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 3620 . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . . 15 (𝑦 = (𝑚𝑣) → (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
2726eqeq2d 2745 . . . . . . . . . . . . . 14 (𝑦 = (𝑚𝑣) → (𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
2827ac6sfi 9317 . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . . . . . 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 24466 . . . . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ 𝑢 = 𝑋) ∧ (𝑚:𝑢𝑋 ∧ ∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑋 = 𝐽)
3732, 36sseqtrd 4035 . . . . . . . . . . . . . . . . 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 7759 . . . . . . . . . . . . . . . . . 18 𝐽 ∈ V
4039elpw2 5339 . . . . . . . . . . . . . . . . 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 1198 . . . . . . . . . . . . . . . . 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 9348 . . . . . . . . . . . . . . . . . 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 4209 . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑚𝑣) → (𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
5150rexrn 7106 . . . . . . . . . . . . . . . . . . 19 (𝑚 Fn 𝑢 → (∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑣𝑢 𝑟 ∈ ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
52 eliun 4999 . . . . . . . . . . . . . . . . . . 19 (𝑟 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
53 eliun 4999 . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . 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 1201 . . . . . . . . . . . . . . . . 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 5062 . . . . . . . . . . . . . . . . . 18 𝑢 = 𝑣𝑢 𝑣
59 iuneq2 5015 . . . . . . . . . . . . . . . . . 18 (∀𝑣𝑢 𝑣 = ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → 𝑣𝑢 𝑣 = 𝑣𝑢 ((𝑚𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
6058, 59eqtrid 2786 . . . . . . . . . . . . . . . . 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 1199 . . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . . 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 5012 . . . . . . . . . . . . . . . 16 (𝑡 = ran 𝑚 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
6564rspceeqv 3644 . . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . . 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 1930 . . . . . . . . . . 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 3153 . . . . . . . . 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 3151 . . . . . . 7 (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑛 ∈ ℕ0𝑡 ∈ (𝒫 𝐽 ∩ Fin)𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
7439pwex 5385 . . . . . . . . 9 𝒫 𝐽 ∈ V
7574inex1 5322 . . . . . . . 8 (𝒫 𝐽 ∩ Fin) ∈ V
76 nn0ennn 14016 . . . . . . . . 9 0 ≈ ℕ
77 nnenom 14017 . . . . . . . . 9 ℕ ≈ ω
7876, 77entri 9046 . . . . . . . 8 0 ≈ ω
79 iuneq1 5012 . . . . . . . . 9 (𝑡 = (𝑚𝑛) → 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
8079eqeq2d 2745 . . . . . . . 8 (𝑡 = (𝑚𝑛) → (𝑋 = 𝑦𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑋 = 𝑦 ∈ (𝑚𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
8175, 78, 80axcc4 10476 . . . . . . 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 4611 . . . . . . . . . 10 (𝑟 ∈ 𝒫 𝐽𝑟𝐽)
84 eqid 2734 . . . . . . . . . . . 12 {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣} = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 𝑣}
85 eqid 2734 . . . . . . . . . . . 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 2734 . . . . . . . . . . . 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 4621 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (CMet‘𝑋) → 𝒫 𝑋 = 𝒫 𝐽)
8988ineq1d 4226 . . . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . . . . . . 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 5044 . . . . . . . . . . . . . . . . . 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 4244 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝒫 𝐽 ∩ Fin) ⊆ 𝒫 𝐽
9796, 88sseqtrrid 4048 . . . . . . . . . . . . . . . . . . . . . . . 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 7103 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚𝑛) ∈ 𝒫 𝑋)
101100elpwid 4613 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚𝑛) ⊆ 𝑋)
102101sselda 3994 . . . . . . . . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑦 → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑚))))
105 oveq2 7438 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
106105oveq2d 7446 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (1 / (2↑𝑚)) = (1 / (2↑𝑛)))
107106oveq2d 7446 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (𝑦(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
108 ovex 7463 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ∈ V
109104, 107, 86, 108ovmpo 7592 . . . . . . . . . . . . . . . . . . . 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 5020 . . . . . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . . . . . 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 2745 . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . 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 5023 . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . . 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 5020 . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . 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 2745 . . . . . . . . . . . . . 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 3234 . . . . . . . . . . . . 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 37806 . . . . . . . . . . 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 3142 . . . . . . . 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 1930 . . . . . 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 2734 . . . . 5 𝐽 = 𝐽
135134iscmp 23411 . . . 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 1086   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wral 3058  wrex 3067  cin 3961  wss 3962  𝒫 cpw 4604   cuni 4911   ciun 4995  {copab 5209  ran crn 5689   Fn wfn 6557  wf 6558  ontowfo 6560  cfv 6562  (class class class)co 7430  cmpo 7432  ωcom 7886  Fincfn 8983  1c1 11153   / cdiv 11917  cn 12263  2c2 12318  0cn0 12523  +crp 13031  cexp 14098  ∞Metcxmet 21366  Metcmet 21367  ballcbl 21368  MetOpencmopn 21371  Topctop 22914  Compccmp 23409  CMetccmet 25301  TotBndctotbnd 37752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-acn 9979  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ico 13389  df-icc 13390  df-fz 13544  df-fl 13828  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-rest 17468  df-topgen 17489  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-top 22915  df-topon 22932  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lm 23252  df-haus 23338  df-cmp 23410  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-cfil 25302  df-cau 25303  df-cmet 25304  df-totbnd 37754
This theorem is referenced by:  rrnheibor  37823
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