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Theorem heibor 36678
Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 36667 and heiborlem1 36668 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 = (MetOpenβ€˜π·)
Assertion
Ref Expression
heibor ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)))
Proof of Theorem heibor
Dummy variables 𝑑 𝑛 𝑦 π‘˜ π‘Ÿ 𝑒 π‘š 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 heibor.1 . . 3 𝐽 = (MetOpenβ€˜π·)
21heibor1 36667 . 2 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ (𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)))
3 cmetmet 24795 . . . 4 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
43adantr 482 . . 3 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
5 metxmet 23832 . . . . . 6 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
61mopntop 23938 . . . . . 6 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
73, 5, 63syl 18 . . . . 5 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐽 ∈ Top)
87adantr 482 . . . 4 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)) β†’ 𝐽 ∈ Top)
9 istotbnd 36626 . . . . . . . . . . . . 13 (𝐷 ∈ (TotBndβ€˜π‘‹) ↔ (𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)π‘Ÿ))))
109simprbi 498 . . . . . . . . . . . 12 (𝐷 ∈ (TotBndβ€˜π‘‹) β†’ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)π‘Ÿ)))
11 2nn 12282 . . . . . . . . . . . . . . 15 2 ∈ β„•
12 nnexpcl 14037 . . . . . . . . . . . . . . 15 ((2 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ (2↑𝑛) ∈ β„•)
1311, 12mpan 689 . . . . . . . . . . . . . 14 (𝑛 ∈ β„•0 β†’ (2↑𝑛) ∈ β„•)
1413nnrpd 13011 . . . . . . . . . . . . 13 (𝑛 ∈ β„•0 β†’ (2↑𝑛) ∈ ℝ+)
1514rpreccld 13023 . . . . . . . . . . . 12 (𝑛 ∈ β„•0 β†’ (1 / (2↑𝑛)) ∈ ℝ+)
16 oveq2 7414 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ = (1 / (2↑𝑛)) β†’ (𝑦(ballβ€˜π·)π‘Ÿ) = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
1716eqeq2d 2744 . . . . . . . . . . . . . . . . 17 (π‘Ÿ = (1 / (2↑𝑛)) β†’ (𝑣 = (𝑦(ballβ€˜π·)π‘Ÿ) ↔ 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
1817rexbidv 3179 . . . . . . . . . . . . . . . 16 (π‘Ÿ = (1 / (2↑𝑛)) β†’ (βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)π‘Ÿ) ↔ βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
1918ralbidv 3178 . . . . . . . . . . . . . . 15 (π‘Ÿ = (1 / (2↑𝑛)) β†’ (βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)π‘Ÿ) ↔ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
2019anbi2d 630 . . . . . . . . . . . . . 14 (π‘Ÿ = (1 / (2↑𝑛)) β†’ ((βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)π‘Ÿ)) ↔ (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))))
2120rexbidv 3179 . . . . . . . . . . . . 13 (π‘Ÿ = (1 / (2↑𝑛)) β†’ (βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)π‘Ÿ)) ↔ βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))))
2221rspccva 3612 . . . . . . . . . . . 12 ((βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)π‘Ÿ)) ∧ (1 / (2↑𝑛)) ∈ ℝ+) β†’ βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
2310, 15, 22syl2an 597 . . . . . . . . . . 11 ((𝐷 ∈ (TotBndβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) β†’ βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
2423expcom 415 . . . . . . . . . 10 (𝑛 ∈ β„•0 β†’ (𝐷 ∈ (TotBndβ€˜π‘‹) β†’ βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))))
2524adantl 483 . . . . . . . . 9 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) β†’ (𝐷 ∈ (TotBndβ€˜π‘‹) β†’ βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))))
26 oveq1 7413 . . . . . . . . . . . . . . 15 (𝑦 = (π‘šβ€˜π‘£) β†’ (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))
2726eqeq2d 2744 . . . . . . . . . . . . . 14 (𝑦 = (π‘šβ€˜π‘£) β†’ (𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) ↔ 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))))
2827ac6sfi 9284 . . . . . . . . . . . . 13 ((𝑒 ∈ Fin ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))) β†’ βˆƒπ‘š(π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))))
2928adantrl 715 . . . . . . . . . . . 12 ((𝑒 ∈ Fin ∧ (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ βˆƒπ‘š(π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))))
3029adantl 483 . . . . . . . . . . 11 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))) β†’ βˆƒπ‘š(π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))))
31 simp3l 1202 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ π‘š:π‘’βŸΆπ‘‹)
3231frnd 6723 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ ran π‘š βŠ† 𝑋)
331mopnuni 23939 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
343, 5, 333syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
3534adantr 482 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) β†’ 𝑋 = βˆͺ 𝐽)
36353ad2ant1 1134 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ 𝑋 = βˆͺ 𝐽)
3732, 36sseqtrd 4022 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ ran π‘š βŠ† βˆͺ 𝐽)
381fvexi 6903 . . . . . . . . . . . . . . . . . . 19 𝐽 ∈ V
3938uniex 7728 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝐽 ∈ V
4039elpw2 5345 . . . . . . . . . . . . . . . . 17 (ran π‘š ∈ 𝒫 βˆͺ 𝐽 ↔ ran π‘š βŠ† βˆͺ 𝐽)
4137, 40sylibr 233 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ ran π‘š ∈ 𝒫 βˆͺ 𝐽)
42 simp2l 1200 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ 𝑒 ∈ Fin)
43 ffn 6715 . . . . . . . . . . . . . . . . . . 19 (π‘š:π‘’βŸΆπ‘‹ β†’ π‘š Fn 𝑒)
44 dffn4 6809 . . . . . . . . . . . . . . . . . . 19 (π‘š Fn 𝑒 ↔ π‘š:𝑒–ontoβ†’ran π‘š)
4543, 44sylib 217 . . . . . . . . . . . . . . . . . 18 (π‘š:π‘’βŸΆπ‘‹ β†’ π‘š:𝑒–ontoβ†’ran π‘š)
46 fofi 9335 . . . . . . . . . . . . . . . . . 18 ((𝑒 ∈ Fin ∧ π‘š:𝑒–ontoβ†’ran π‘š) β†’ ran π‘š ∈ Fin)
4745, 46sylan2 594 . . . . . . . . . . . . . . . . 17 ((𝑒 ∈ Fin ∧ π‘š:π‘’βŸΆπ‘‹) β†’ ran π‘š ∈ Fin)
4842, 31, 47syl2anc 585 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ ran π‘š ∈ Fin)
4941, 48elind 4194 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ ran π‘š ∈ (𝒫 βˆͺ 𝐽 ∩ Fin))
5026eleq2d 2820 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (π‘šβ€˜π‘£) β†’ (π‘Ÿ ∈ (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) ↔ π‘Ÿ ∈ ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))))
5150rexrn 7086 . . . . . . . . . . . . . . . . . . 19 (π‘š Fn 𝑒 β†’ (βˆƒπ‘¦ ∈ ran π‘š π‘Ÿ ∈ (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) ↔ βˆƒπ‘£ ∈ 𝑒 π‘Ÿ ∈ ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))))
52 eliun 5001 . . . . . . . . . . . . . . . . . . 19 (π‘Ÿ ∈ βˆͺ 𝑦 ∈ ran π‘š(𝑦(ballβ€˜π·)(1 / (2↑𝑛))) ↔ βˆƒπ‘¦ ∈ ran π‘š π‘Ÿ ∈ (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
53 eliun 5001 . . . . . . . . . . . . . . . . . . 19 (π‘Ÿ ∈ βˆͺ 𝑣 ∈ 𝑒 ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))) ↔ βˆƒπ‘£ ∈ 𝑒 π‘Ÿ ∈ ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))
5451, 52, 533bitr4g 314 . . . . . . . . . . . . . . . . . 18 (π‘š Fn 𝑒 β†’ (π‘Ÿ ∈ βˆͺ 𝑦 ∈ ran π‘š(𝑦(ballβ€˜π·)(1 / (2↑𝑛))) ↔ π‘Ÿ ∈ βˆͺ 𝑣 ∈ 𝑒 ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))))
5554eqrdv 2731 . . . . . . . . . . . . . . . . 17 (π‘š Fn 𝑒 β†’ βˆͺ 𝑦 ∈ ran π‘š(𝑦(ballβ€˜π·)(1 / (2↑𝑛))) = βˆͺ 𝑣 ∈ 𝑒 ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))
5631, 43, 553syl 18 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ βˆͺ 𝑦 ∈ ran π‘š(𝑦(ballβ€˜π·)(1 / (2↑𝑛))) = βˆͺ 𝑣 ∈ 𝑒 ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))
57 simp3r 1203 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))
58 uniiun 5061 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝑒 = βˆͺ 𝑣 ∈ 𝑒 𝑣
59 iuneq2 5016 . . . . . . . . . . . . . . . . . 18 (βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))) β†’ βˆͺ 𝑣 ∈ 𝑒 𝑣 = βˆͺ 𝑣 ∈ 𝑒 ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))
6058, 59eqtrid 2785 . . . . . . . . . . . . . . . . 17 (βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))) β†’ βˆͺ 𝑒 = βˆͺ 𝑣 ∈ 𝑒 ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))
6157, 60syl 17 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ βˆͺ 𝑒 = βˆͺ 𝑣 ∈ 𝑒 ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))
62 simp2r 1201 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ βˆͺ 𝑒 = 𝑋)
6356, 61, 623eqtr2rd 2780 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ 𝑋 = βˆͺ 𝑦 ∈ ran π‘š(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
64 iuneq1 5013 . . . . . . . . . . . . . . . 16 (𝑑 = ran π‘š β†’ βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) = βˆͺ 𝑦 ∈ ran π‘š(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
6564rspceeqv 3633 . . . . . . . . . . . . . . 15 ((ran π‘š ∈ (𝒫 βˆͺ 𝐽 ∩ Fin) ∧ 𝑋 = βˆͺ 𝑦 ∈ ran π‘š(𝑦(ballβ€˜π·)(1 / (2↑𝑛)))) β†’ βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
6649, 63, 65syl2anc 585 . . . . . . . . . . . . . 14 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋) ∧ (π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
67663expia 1122 . . . . . . . . . . . . 13 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ βˆͺ 𝑒 = 𝑋)) β†’ ((π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))) β†’ βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
6867adantrrr 724 . . . . . . . . . . . 12 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))) β†’ ((π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))) β†’ βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
6968exlimdv 1937 . . . . . . . . . . 11 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))) β†’ (βˆƒπ‘š(π‘š:π‘’βŸΆπ‘‹ ∧ βˆ€π‘£ ∈ 𝑒 𝑣 = ((π‘šβ€˜π‘£)(ballβ€˜π·)(1 / (2↑𝑛)))) β†’ βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
7030, 69mpd 15 . . . . . . . . . 10 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) ∧ (𝑒 ∈ Fin ∧ (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))) β†’ βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
7170rexlimdvaa 3157 . . . . . . . . 9 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) β†’ (βˆƒπ‘’ ∈ Fin (βˆͺ 𝑒 = 𝑋 ∧ βˆ€π‘£ ∈ 𝑒 βˆƒπ‘¦ ∈ 𝑋 𝑣 = (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))) β†’ βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
7225, 71syld 47 . . . . . . . 8 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑛 ∈ β„•0) β†’ (𝐷 ∈ (TotBndβ€˜π‘‹) β†’ βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
7372ralrimdva 3155 . . . . . . 7 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ (𝐷 ∈ (TotBndβ€˜π‘‹) β†’ βˆ€π‘› ∈ β„•0 βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
7439pwex 5378 . . . . . . . . 9 𝒫 βˆͺ 𝐽 ∈ V
7574inex1 5317 . . . . . . . 8 (𝒫 βˆͺ 𝐽 ∩ Fin) ∈ V
76 nn0ennn 13941 . . . . . . . . 9 β„•0 β‰ˆ β„•
77 nnenom 13942 . . . . . . . . 9 β„• β‰ˆ Ο‰
7876, 77entri 9001 . . . . . . . 8 β„•0 β‰ˆ Ο‰
79 iuneq1 5013 . . . . . . . . 9 (𝑑 = (π‘šβ€˜π‘›) β†’ βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
8079eqeq2d 2744 . . . . . . . 8 (𝑑 = (π‘šβ€˜π‘›) β†’ (𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) ↔ 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
8175, 78, 80axcc4 10431 . . . . . . 7 (βˆ€π‘› ∈ β„•0 βˆƒπ‘‘ ∈ (𝒫 βˆͺ 𝐽 ∩ Fin)𝑋 = βˆͺ 𝑦 ∈ 𝑑 (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) β†’ βˆƒπ‘š(π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
8273, 81syl6 35 . . . . . 6 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ (𝐷 ∈ (TotBndβ€˜π‘‹) β†’ βˆƒπ‘š(π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))))
83 elpwi 4609 . . . . . . . . . 10 (π‘Ÿ ∈ 𝒫 𝐽 β†’ π‘Ÿ βŠ† 𝐽)
84 eqid 2733 . . . . . . . . . . . 12 {𝑒 ∣ Β¬ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)𝑒 βŠ† βˆͺ 𝑣} = {𝑒 ∣ Β¬ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)𝑒 βŠ† βˆͺ 𝑣}
85 eqid 2733 . . . . . . . . . . . 12 {βŸ¨π‘‘, π‘˜βŸ© ∣ (π‘˜ ∈ β„•0 ∧ 𝑑 ∈ (π‘šβ€˜π‘˜) ∧ (𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))π‘˜) ∈ {𝑒 ∣ Β¬ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)𝑒 βŠ† βˆͺ 𝑣})} = {βŸ¨π‘‘, π‘˜βŸ© ∣ (π‘˜ ∈ β„•0 ∧ 𝑑 ∈ (π‘šβ€˜π‘˜) ∧ (𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))π‘˜) ∈ {𝑒 ∣ Β¬ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)𝑒 βŠ† βˆͺ 𝑣})}
86 eqid 2733 . . . . . . . . . . . 12 (𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š)))) = (𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))
87 simpl 484 . . . . . . . . . . . 12 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
8834pweqd 4619 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
8988ineq1d 4211 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ (𝒫 𝑋 ∩ Fin) = (𝒫 βˆͺ 𝐽 ∩ Fin))
9089feq3d 6702 . . . . . . . . . . . . . 14 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ (π‘š:β„•0⟢(𝒫 𝑋 ∩ Fin) ↔ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)))
9190biimpar 479 . . . . . . . . . . . . 13 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) β†’ π‘š:β„•0⟢(𝒫 𝑋 ∩ Fin))
9291adantrr 716 . . . . . . . . . . . 12 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ π‘š:β„•0⟢(𝒫 𝑋 ∩ Fin))
93 oveq1 7413 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑦 β†’ (𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = (𝑦(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛))
9493cbviunv 5043 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛)
95 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) β†’ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin))
96 inss1 4228 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝒫 βˆͺ 𝐽 ∩ Fin) βŠ† 𝒫 βˆͺ 𝐽
9796, 88sseqtrrid 4035 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ (𝒫 βˆͺ 𝐽 ∩ Fin) βŠ† 𝒫 𝑋)
98 fss 6732 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ (𝒫 βˆͺ 𝐽 ∩ Fin) βŠ† 𝒫 𝑋) β†’ π‘š:β„•0βŸΆπ’« 𝑋)
9995, 97, 98syl2anr 598 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) β†’ π‘š:β„•0βŸΆπ’« 𝑋)
10099ffvelcdmda 7084 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) β†’ (π‘šβ€˜π‘›) ∈ 𝒫 𝑋)
101100elpwid 4611 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) β†’ (π‘šβ€˜π‘›) βŠ† 𝑋)
102101sselda 3982 . . . . . . . . . . . . . . . . . . . 20 ((((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) ∧ 𝑦 ∈ (π‘šβ€˜π‘›)) β†’ 𝑦 ∈ 𝑋)
103 simplr 768 . . . . . . . . . . . . . . . . . . . 20 ((((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) ∧ 𝑦 ∈ (π‘šβ€˜π‘›)) β†’ 𝑛 ∈ β„•0)
104 oveq1 7413 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑦 β†’ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))) = (𝑦(ballβ€˜π·)(1 / (2β†‘π‘š))))
105 oveq2 7414 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘š = 𝑛 β†’ (2β†‘π‘š) = (2↑𝑛))
106105oveq2d 7422 . . . . . . . . . . . . . . . . . . . . . 22 (π‘š = 𝑛 β†’ (1 / (2β†‘π‘š)) = (1 / (2↑𝑛)))
107106oveq2d 7422 . . . . . . . . . . . . . . . . . . . . 21 (π‘š = 𝑛 β†’ (𝑦(ballβ€˜π·)(1 / (2β†‘π‘š))) = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
108 ovex 7439 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) ∈ V
109104, 107, 86, 108ovmpo 7565 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ 𝑋 ∧ 𝑛 ∈ β„•0) β†’ (𝑦(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
110102, 103, 109syl2anc 585 . . . . . . . . . . . . . . . . . . 19 ((((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) ∧ 𝑦 ∈ (π‘šβ€˜π‘›)) β†’ (𝑦(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
111110iuneq2dv 5021 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) β†’ βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
11294, 111eqtrid 2785 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) β†’ βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
113112eqeq2d 2744 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) β†’ (𝑋 = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) ↔ 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛)))))
114113biimprd 247 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) β†’ (𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))) β†’ 𝑋 = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛)))
115114ralimdva 3168 . . . . . . . . . . . . . 14 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) β†’ (βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))) β†’ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛)))
116115impr 456 . . . . . . . . . . . . 13 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛))
117 fveq2 6889 . . . . . . . . . . . . . . . . 17 (𝑛 = π‘˜ β†’ (π‘šβ€˜π‘›) = (π‘šβ€˜π‘˜))
118117iuneq1d 5024 . . . . . . . . . . . . . . . 16 (𝑛 = π‘˜ β†’ βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘˜)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛))
119 simpl 484 . . . . . . . . . . . . . . . . . 18 ((𝑛 = π‘˜ ∧ 𝑑 ∈ (π‘šβ€˜π‘˜)) β†’ 𝑛 = π‘˜)
120119oveq2d 7422 . . . . . . . . . . . . . . . . 17 ((𝑛 = π‘˜ ∧ 𝑑 ∈ (π‘šβ€˜π‘˜)) β†’ (𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = (𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))π‘˜))
121120iuneq2dv 5021 . . . . . . . . . . . . . . . 16 (𝑛 = π‘˜ β†’ βˆͺ 𝑑 ∈ (π‘šβ€˜π‘˜)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘˜)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))π‘˜))
122118, 121eqtrd 2773 . . . . . . . . . . . . . . 15 (𝑛 = π‘˜ β†’ βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘˜)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))π‘˜))
123122eqeq2d 2744 . . . . . . . . . . . . . 14 (𝑛 = π‘˜ β†’ (𝑋 = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) ↔ 𝑋 = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘˜)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))π‘˜)))
124123cbvralvw 3235 . . . . . . . . . . . . 13 (βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘›)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))𝑛) ↔ βˆ€π‘˜ ∈ β„•0 𝑋 = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘˜)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))π‘˜))
125116, 124sylib 217 . . . . . . . . . . . 12 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ βˆ€π‘˜ ∈ β„•0 𝑋 = βˆͺ 𝑑 ∈ (π‘šβ€˜π‘˜)(𝑑(𝑧 ∈ 𝑋, π‘š ∈ β„•0 ↦ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))))π‘˜))
1261, 84, 85, 86, 87, 92, 125heiborlem10 36677 . . . . . . . . . . 11 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))) ∧ (π‘Ÿ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘Ÿ)) β†’ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑣)
127126exp32 422 . . . . . . . . . 10 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ (π‘Ÿ βŠ† 𝐽 β†’ (βˆͺ 𝐽 = βˆͺ π‘Ÿ β†’ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑣)))
12883, 127syl5 34 . . . . . . . . 9 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ (π‘Ÿ ∈ 𝒫 𝐽 β†’ (βˆͺ 𝐽 = βˆͺ π‘Ÿ β†’ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑣)))
129128ralrimiv 3146 . . . . . . . 8 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ (π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛))))) β†’ βˆ€π‘Ÿ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ π‘Ÿ β†’ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑣))
130129ex 414 . . . . . . 7 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ ((π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛)))) β†’ βˆ€π‘Ÿ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ π‘Ÿ β†’ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑣)))
131130exlimdv 1937 . . . . . 6 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ (βˆƒπ‘š(π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin) ∧ βˆ€π‘› ∈ β„•0 𝑋 = βˆͺ 𝑦 ∈ (π‘šβ€˜π‘›)(𝑦(ballβ€˜π·)(1 / (2↑𝑛)))) β†’ βˆ€π‘Ÿ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ π‘Ÿ β†’ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑣)))
13282, 131syld 47 . . . . 5 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ (𝐷 ∈ (TotBndβ€˜π‘‹) β†’ βˆ€π‘Ÿ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ π‘Ÿ β†’ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑣)))
133132imp 408 . . . 4 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)) β†’ βˆ€π‘Ÿ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ π‘Ÿ β†’ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑣))
134 eqid 2733 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
135134iscmp 22884 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ βˆ€π‘Ÿ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ π‘Ÿ β†’ βˆƒπ‘£ ∈ (𝒫 π‘Ÿ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑣)))
1368, 133, 135sylanbrc 584 . . 3 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)) β†’ 𝐽 ∈ Comp)
1374, 136jca 513 . 2 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)) β†’ (𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp))
1382, 137impbii 208 1 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908  βˆͺ ciun 4997  {copab 5210  ran crn 5677   Fn wfn 6536  βŸΆwf 6537  β€“ontoβ†’wfo 6539  β€˜cfv 6541  (class class class)co 7406   ∈ cmpo 7408  Ο‰com 7852  Fincfn 8936  1c1 11108   / cdiv 11868  β„•cn 12209  2c2 12264  β„•0cn0 12469  β„+crp 12971  β†‘cexp 14024  βˆžMetcxmet 20922  Metcmet 20923  ballcbl 20924  MetOpencmopn 20927  Topctop 22387  Compccmp 22882  CMetccmet 24763  TotBndctotbnd 36623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cc 10427  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-oadd 8467  df-omul 8468  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9502  df-card 9931  df-acn 9934  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-ico 13327  df-icc 13328  df-fz 13482  df-fl 13754  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-rlim 15430  df-rest 17365  df-topgen 17386  df-psmet 20929  df-xmet 20930  df-met 20931  df-bl 20932  df-mopn 20933  df-fbas 20934  df-fg 20935  df-top 22388  df-topon 22405  df-bases 22441  df-cld 22515  df-ntr 22516  df-cls 22517  df-nei 22594  df-lm 22725  df-haus 22811  df-cmp 22883  df-fil 23342  df-fm 23434  df-flim 23435  df-flf 23436  df-cfil 24764  df-cau 24765  df-cmet 24766  df-totbnd 36625
This theorem is referenced by:  rrnheibor  36694
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