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Theorem heibor1 36666
Description: One half of heibor 36677, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 24827 and total boundedness here, which follows trivially from the fact that the set of all π‘Ÿ-balls is an open cover of 𝑋, so finitely many cover 𝑋. (Contributed by Jeff Madsen, 16-Jan-2014.)
Hypothesis
Ref Expression
heibor.1 𝐽 = (MetOpenβ€˜π·)
Assertion
Ref Expression
heibor1 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ (𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)))
Proof of Theorem heibor1
Dummy variables π‘₯ 𝑦 π‘Ÿ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 heibor.1 . . . . . 6 𝐽 = (MetOpenβ€˜π·)
2 simpll 765 . . . . . 6 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ (π‘₯ ∈ (Cauβ€˜π·) ∧ π‘₯:β„•βŸΆπ‘‹)) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
3 simplr 767 . . . . . 6 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ (π‘₯ ∈ (Cauβ€˜π·) ∧ π‘₯:β„•βŸΆπ‘‹)) β†’ 𝐽 ∈ Comp)
4 simprl 769 . . . . . 6 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ (π‘₯ ∈ (Cauβ€˜π·) ∧ π‘₯:β„•βŸΆπ‘‹)) β†’ π‘₯ ∈ (Cauβ€˜π·))
5 simprr 771 . . . . . 6 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ (π‘₯ ∈ (Cauβ€˜π·) ∧ π‘₯:β„•βŸΆπ‘‹)) β†’ π‘₯:β„•βŸΆπ‘‹)
61, 2, 3, 4, 5heibor1lem 36665 . . . . 5 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ (π‘₯ ∈ (Cauβ€˜π·) ∧ π‘₯:β„•βŸΆπ‘‹)) β†’ π‘₯ ∈ dom (β‡π‘‘β€˜π½))
76expr 457 . . . 4 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘₯ ∈ (Cauβ€˜π·)) β†’ (π‘₯:β„•βŸΆπ‘‹ β†’ π‘₯ ∈ dom (β‡π‘‘β€˜π½)))
87ralrimiva 3146 . . 3 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ βˆ€π‘₯ ∈ (Cauβ€˜π·)(π‘₯:β„•βŸΆπ‘‹ β†’ π‘₯ ∈ dom (β‡π‘‘β€˜π½)))
9 nnuz 12861 . . . 4 β„• = (β„€β‰₯β€˜1)
10 1zzd 12589 . . . 4 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ 1 ∈ β„€)
11 simpl 483 . . . 4 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
129, 1, 10, 11iscmet3 24801 . . 3 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ (𝐷 ∈ (CMetβ€˜π‘‹) ↔ βˆ€π‘₯ ∈ (Cauβ€˜π·)(π‘₯:β„•βŸΆπ‘‹ β†’ π‘₯ ∈ dom (β‡π‘‘β€˜π½))))
138, 12mpbird 256 . 2 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
14 simplr 767 . . . . . . 7 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ 𝐽 ∈ Comp)
15 metxmet 23831 . . . . . . . . . . . . . 14 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
16 id 22 . . . . . . . . . . . . . 14 (𝑧 ∈ 𝑋 β†’ 𝑧 ∈ 𝑋)
17 rpxr 12979 . . . . . . . . . . . . . 14 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
181blopn 24000 . . . . . . . . . . . . . 14 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑧(ballβ€˜π·)π‘Ÿ) ∈ 𝐽)
1915, 16, 17, 18syl3an 1160 . . . . . . . . . . . . 13 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ (𝑧(ballβ€˜π·)π‘Ÿ) ∈ 𝐽)
20193com23 1126 . . . . . . . . . . . 12 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) β†’ (𝑧(ballβ€˜π·)π‘Ÿ) ∈ 𝐽)
21203expa 1118 . . . . . . . . . . 11 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) β†’ (𝑧(ballβ€˜π·)π‘Ÿ) ∈ 𝐽)
22 eleq1a 2828 . . . . . . . . . . 11 ((𝑧(ballβ€˜π·)π‘Ÿ) ∈ 𝐽 β†’ (𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ) β†’ 𝑦 ∈ 𝐽))
2321, 22syl 17 . . . . . . . . . 10 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) β†’ (𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ) β†’ 𝑦 ∈ 𝐽))
2423rexlimdva 3155 . . . . . . . . 9 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ+) β†’ (βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ) β†’ 𝑦 ∈ 𝐽))
2524adantlr 713 . . . . . . . 8 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ (βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ) β†’ 𝑦 ∈ 𝐽))
2625abssdv 4064 . . . . . . 7 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} βŠ† 𝐽)
2715ad2antrr 724 . . . . . . . . . 10 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
281mopnuni 23938 . . . . . . . . . 10 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2927, 28syl 17 . . . . . . . . 9 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑋 = βˆͺ 𝐽)
30 blcntr 23910 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑧 ∈ (𝑧(ballβ€˜π·)π‘Ÿ))
3115, 30syl3an1 1163 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑧 ∈ (𝑧(ballβ€˜π·)π‘Ÿ))
32313com23 1126 . . . . . . . . . . . . . 14 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) β†’ 𝑧 ∈ (𝑧(ballβ€˜π·)π‘Ÿ))
33323expa 1118 . . . . . . . . . . . . 13 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) β†’ 𝑧 ∈ (𝑧(ballβ€˜π·)π‘Ÿ))
34 ovex 7438 . . . . . . . . . . . . . . 15 (𝑧(ballβ€˜π·)π‘Ÿ) ∈ V
3534elabrex 7238 . . . . . . . . . . . . . 14 (𝑧 ∈ 𝑋 β†’ (𝑧(ballβ€˜π·)π‘Ÿ) ∈ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
3635adantl 482 . . . . . . . . . . . . 13 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) β†’ (𝑧(ballβ€˜π·)π‘Ÿ) ∈ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
37 elunii 4912 . . . . . . . . . . . . 13 ((𝑧 ∈ (𝑧(ballβ€˜π·)π‘Ÿ) ∧ (𝑧(ballβ€˜π·)π‘Ÿ) ∈ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}) β†’ 𝑧 ∈ βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
3833, 36, 37syl2anc 584 . . . . . . . . . . . 12 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) β†’ 𝑧 ∈ βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
3938ralrimiva 3146 . . . . . . . . . . 11 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ π‘Ÿ ∈ ℝ+) β†’ βˆ€π‘§ ∈ 𝑋 𝑧 ∈ βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
4039adantlr 713 . . . . . . . . . 10 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ βˆ€π‘§ ∈ 𝑋 𝑧 ∈ βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
41 nfcv 2903 . . . . . . . . . . 11 Ⅎ𝑧𝑋
42 nfre1 3282 . . . . . . . . . . . . 13 β„²π‘§βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)
4342nfab 2909 . . . . . . . . . . . 12 Ⅎ𝑧{𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}
4443nfuni 4914 . . . . . . . . . . 11 Ⅎ𝑧βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}
4541, 44dfss3f 3972 . . . . . . . . . 10 (𝑋 βŠ† βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ↔ βˆ€π‘§ ∈ 𝑋 𝑧 ∈ βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
4640, 45sylibr 233 . . . . . . . . 9 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑋 βŠ† βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
4729, 46eqsstrrd 4020 . . . . . . . 8 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ βˆͺ 𝐽 βŠ† βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
4826unissd 4917 . . . . . . . 8 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} βŠ† βˆͺ 𝐽)
4947, 48eqssd 3998 . . . . . . 7 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ βˆͺ 𝐽 = βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
50 eqid 2732 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
5150cmpcov 22884 . . . . . . 7 ((𝐽 ∈ Comp ∧ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}) β†’ βˆƒπ‘₯ ∈ (𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∩ Fin)βˆͺ 𝐽 = βˆͺ π‘₯)
5214, 26, 49, 51syl3anc 1371 . . . . . 6 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ βˆƒπ‘₯ ∈ (𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∩ Fin)βˆͺ 𝐽 = βˆͺ π‘₯)
53 elin 3963 . . . . . . . . . 10 (π‘₯ ∈ (𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∩ Fin) ↔ (π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ π‘₯ ∈ Fin))
54 ancom 461 . . . . . . . . . 10 ((π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ π‘₯ ∈ Fin) ↔ (π‘₯ ∈ Fin ∧ π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}))
5553, 54bitri 274 . . . . . . . . 9 (π‘₯ ∈ (𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∩ Fin) ↔ (π‘₯ ∈ Fin ∧ π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}))
5655anbi1i 624 . . . . . . . 8 ((π‘₯ ∈ (𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∩ Fin) ∧ βˆͺ 𝐽 = βˆͺ π‘₯) ↔ ((π‘₯ ∈ Fin ∧ π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}) ∧ βˆͺ 𝐽 = βˆͺ π‘₯))
57 anass 469 . . . . . . . 8 (((π‘₯ ∈ Fin ∧ π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}) ∧ βˆͺ 𝐽 = βˆͺ π‘₯) ↔ (π‘₯ ∈ Fin ∧ (π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ βˆͺ 𝐽 = βˆͺ π‘₯)))
5856, 57bitri 274 . . . . . . 7 ((π‘₯ ∈ (𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∩ Fin) ∧ βˆͺ 𝐽 = βˆͺ π‘₯) ↔ (π‘₯ ∈ Fin ∧ (π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ βˆͺ 𝐽 = βˆͺ π‘₯)))
5958rexbii2 3090 . . . . . 6 (βˆƒπ‘₯ ∈ (𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∩ Fin)βˆͺ 𝐽 = βˆͺ π‘₯ ↔ βˆƒπ‘₯ ∈ Fin (π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ βˆͺ 𝐽 = βˆͺ π‘₯))
6052, 59sylib 217 . . . . 5 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ βˆƒπ‘₯ ∈ Fin (π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ βˆͺ 𝐽 = βˆͺ π‘₯))
61 ancom 461 . . . . . . . 8 ((π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ βˆͺ 𝐽 = βˆͺ π‘₯) ↔ (βˆͺ 𝐽 = βˆͺ π‘₯ ∧ π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}))
62 eqcom 2739 . . . . . . . . . 10 (βˆͺ π‘₯ = 𝑋 ↔ 𝑋 = βˆͺ π‘₯)
6329eqeq1d 2734 . . . . . . . . . 10 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ (𝑋 = βˆͺ π‘₯ ↔ βˆͺ 𝐽 = βˆͺ π‘₯))
6462, 63bitr2id 283 . . . . . . . . 9 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ (βˆͺ 𝐽 = βˆͺ π‘₯ ↔ βˆͺ π‘₯ = 𝑋))
6564anbi1d 630 . . . . . . . 8 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ ((βˆͺ 𝐽 = βˆͺ π‘₯ ∧ π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}) ↔ (βˆͺ π‘₯ = 𝑋 ∧ π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})))
6661, 65bitrid 282 . . . . . . 7 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ ((π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ βˆͺ 𝐽 = βˆͺ π‘₯) ↔ (βˆͺ π‘₯ = 𝑋 ∧ π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})))
67 elpwi 4608 . . . . . . . . 9 (π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} β†’ π‘₯ βŠ† {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
68 ssabral 4058 . . . . . . . . 9 (π‘₯ βŠ† {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ↔ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ))
6967, 68sylib 217 . . . . . . . 8 (π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} β†’ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ))
7069anim2i 617 . . . . . . 7 ((βˆͺ π‘₯ = 𝑋 ∧ π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}) β†’ (βˆͺ π‘₯ = 𝑋 ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)))
7166, 70syl6bi 252 . . . . . 6 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ ((π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ βˆͺ 𝐽 = βˆͺ π‘₯) β†’ (βˆͺ π‘₯ = 𝑋 ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ))))
7271reximdv 3170 . . . . 5 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ (βˆƒπ‘₯ ∈ Fin (π‘₯ ∈ 𝒫 {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)} ∧ βˆͺ 𝐽 = βˆͺ π‘₯) β†’ βˆƒπ‘₯ ∈ Fin (βˆͺ π‘₯ = 𝑋 ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ))))
7360, 72mpd 15 . . . 4 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ βˆƒπ‘₯ ∈ Fin (βˆͺ π‘₯ = 𝑋 ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)))
7473ralrimiva 3146 . . 3 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘₯ ∈ Fin (βˆͺ π‘₯ = 𝑋 ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)))
75 istotbnd 36625 . . 3 (𝐷 ∈ (TotBndβ€˜π‘‹) ↔ (𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘₯ ∈ Fin (βˆͺ π‘₯ = 𝑋 ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ))))
7611, 74, 75sylanbrc 583 . 2 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ 𝐷 ∈ (TotBndβ€˜π‘‹))
7713, 76jca 512 1 ((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) β†’ (𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝐷 ∈ (TotBndβ€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907  dom cdm 5675  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  Fincfn 8935  1c1 11107  β„*cxr 11243  β„•cn 12208  β„+crp 12970  βˆžMetcxmet 20921  Metcmet 20922  ballcbl 20923  MetOpencmopn 20926  β‡π‘‘clm 22721  Compccmp 22881  Cauccau 24761  CMetccmet 24762  TotBndctotbnd 36622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cc 10426  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-omul 8467  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-card 9930  df-acn 9933  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ico 13326  df-fz 13481  df-fl 13753  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-rlim 15429  df-rest 17364  df-topgen 17385  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-fbas 20933  df-fg 20934  df-top 22387  df-topon 22404  df-bases 22440  df-cld 22514  df-ntr 22515  df-cls 22516  df-nei 22593  df-lm 22724  df-cmp 22882  df-fil 23341  df-fm 23433  df-flim 23434  df-flf 23435  df-cfil 24763  df-cau 24764  df-cmet 24765  df-totbnd 36624
This theorem is referenced by:  heibor  36677
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