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Theorem heibor 36737
Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 36726 and heiborlem1 36727 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 36726 . 2 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ (𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)))
3 cmetmet 24803 . . . 4 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
43adantr 482 . . 3 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
5 metxmet 23840 . . . . . 6 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
61mopntop 23946 . . . . . 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 36685 . . . . . . . . . . . . 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 12285 . . . . . . . . . . . . . . 15 2 ∈ β„•
12 nnexpcl 14040 . . . . . . . . . . . . . . 15 ((2 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ (2↑𝑛) ∈ β„•)
1311, 12mpan 689 . . . . . . . . . . . . . 14 (𝑛 ∈ β„•0 β†’ (2↑𝑛) ∈ β„•)
1413nnrpd 13014 . . . . . . . . . . . . 13 (𝑛 ∈ β„•0 β†’ (2↑𝑛) ∈ ℝ+)
1514rpreccld 13026 . . . . . . . . . . . 12 (𝑛 ∈ β„•0 β†’ (1 / (2↑𝑛)) ∈ ℝ+)
16 oveq2 7417 . . . . . . . . . . . . . . . . . 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 7416 . . . . . . . . . . . . . . 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 9287 . . . . . . . . . . . . 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 6726 . . . . . . . . . . . . . . . . . 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 23947 . . . . . . . . . . . . . . . . . . . . 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 4023 . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . 19 𝐽 ∈ V
3938uniex 7731 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝐽 ∈ V
4039elpw2 5346 . . . . . . . . . . . . . . . . 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 6718 . . . . . . . . . . . . . . . . . . 19 (π‘š:π‘’βŸΆπ‘‹ β†’ π‘š Fn 𝑒)
44 dffn4 6812 . . . . . . . . . . . . . . . . . . 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 9338 . . . . . . . . . . . . . . . . . 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 4195 . . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . . . . . . 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 5002 . . . . . . . . . . . . . . . . . . 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 5002 . . . . . . . . . . . . . . . . . . 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 5062 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝑒 = βˆͺ 𝑣 ∈ 𝑒 𝑣
59 iuneq2 5017 . . . . . . . . . . . . . . . . . 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 5014 . . . . . . . . . . . . . . . 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 3634 . . . . . . . . . . . . . . 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 5379 . . . . . . . . 9 𝒫 βˆͺ 𝐽 ∈ V
7574inex1 5318 . . . . . . . 8 (𝒫 βˆͺ 𝐽 ∩ Fin) ∈ V
76 nn0ennn 13944 . . . . . . . . 9 β„•0 β‰ˆ β„•
77 nnenom 13945 . . . . . . . . 9 β„• β‰ˆ Ο‰
7876, 77entri 9004 . . . . . . . 8 β„•0 β‰ˆ Ο‰
79 iuneq1 5014 . . . . . . . . 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 10434 . . . . . . 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 4610 . . . . . . . . . 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 4620 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
8988ineq1d 4212 . . . . . . . . . . . . . . 15 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ (𝒫 𝑋 ∩ Fin) = (𝒫 βˆͺ 𝐽 ∩ Fin))
9089feq3d 6705 . . . . . . . . . . . . . 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 7416 . . . . . . . . . . . . . . . . . . 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 5044 . . . . . . . . . . . . . . . . . 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 4229 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝒫 βˆͺ 𝐽 ∩ Fin) βŠ† 𝒫 βˆͺ 𝐽
9796, 88sseqtrrid 4036 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ (𝒫 βˆͺ 𝐽 ∩ Fin) βŠ† 𝒫 𝑋)
98 fss 6735 . . . . . . . . . . . . . . . . . . . . . . . 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 7087 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) β†’ (π‘šβ€˜π‘›) ∈ 𝒫 𝑋)
101100elpwid 4612 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ π‘š:β„•0⟢(𝒫 βˆͺ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ β„•0) β†’ (π‘šβ€˜π‘›) βŠ† 𝑋)
102101sselda 3983 . . . . . . . . . . . . . . . . . . . 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 7416 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑦 β†’ (𝑧(ballβ€˜π·)(1 / (2β†‘π‘š))) = (𝑦(ballβ€˜π·)(1 / (2β†‘π‘š))))
105 oveq2 7417 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘š = 𝑛 β†’ (2β†‘π‘š) = (2↑𝑛))
106105oveq2d 7425 . . . . . . . . . . . . . . . . . . . . . 22 (π‘š = 𝑛 β†’ (1 / (2β†‘π‘š)) = (1 / (2↑𝑛)))
107106oveq2d 7425 . . . . . . . . . . . . . . . . . . . . 21 (π‘š = 𝑛 β†’ (𝑦(ballβ€˜π·)(1 / (2β†‘π‘š))) = (𝑦(ballβ€˜π·)(1 / (2↑𝑛))))
108 ovex 7442 . . . . . . . . . . . . . . . . . . . . 21 (𝑦(ballβ€˜π·)(1 / (2↑𝑛))) ∈ V
109104, 107, 86, 108ovmpo 7568 . . . . . . . . . . . . . . . . . . . 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 5022 . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . 17 (𝑛 = π‘˜ β†’ (π‘šβ€˜π‘›) = (π‘šβ€˜π‘˜))
118117iuneq1d 5025 . . . . . . . . . . . . . . . 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 7425 . . . . . . . . . . . . . . . . 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 5022 . . . . . . . . . . . . . . . 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 36736 . . . . . . . . . . 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 22892 . . . 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 3948   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909  βˆͺ ciun 4998  {copab 5211  ran crn 5678   Fn wfn 6539  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  Ο‰com 7855  Fincfn 8939  1c1 11111   / cdiv 11871  β„•cn 12212  2c2 12267  β„•0cn0 12472  β„+crp 12974  β†‘cexp 14027  βˆžMetcxmet 20929  Metcmet 20930  ballcbl 20931  MetOpencmopn 20934  Topctop 22395  Compccmp 22890  CMetccmet 24771  TotBndctotbnd 36682
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cc 10430  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-acn 9937  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ico 13330  df-icc 13331  df-fz 13485  df-fl 13757  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-rlim 15433  df-rest 17368  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-lm 22733  df-haus 22819  df-cmp 22891  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444  df-cfil 24772  df-cau 24773  df-cmet 24774  df-totbnd 36684
This theorem is referenced by:  rrnheibor  36753
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