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Theorem heibor1 36981
Description: One half of heibor 36992, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 25060 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 36980 . . . . 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 12869 . . . 4 β„• = (β„€β‰₯β€˜1)
10 1zzd 12597 . . . 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 25034 . . 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 24060 . . . . . . . . . . . . . 14 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
16 id 22 . . . . . . . . . . . . . 14 (𝑧 ∈ 𝑋 β†’ 𝑧 ∈ 𝑋)
17 rpxr 12987 . . . . . . . . . . . . . 14 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
181blopn 24229 . . . . . . . . . . . . . 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 4065 . . . . . . 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 24167 . . . . . . . . . 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 24139 . . . . . . . . . . . . . . . 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 7444 . . . . . . . . . . . . . . 15 (𝑧(ballβ€˜π·)π‘Ÿ) ∈ V
3534elabrex 7244 . . . . . . . . . . . . . 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 4913 . . . . . . . . . . . . 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 4915 . . . . . . . . . . 11 Ⅎ𝑧βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)}
4541, 44dfss3f 3973 . . . . . . . . . 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 4021 . . . . . . . 8 (((𝐷 ∈ (Metβ€˜π‘‹) ∧ 𝐽 ∈ Comp) ∧ π‘Ÿ ∈ ℝ+) β†’ βˆͺ 𝐽 βŠ† βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ 𝑋 𝑦 = (𝑧(ballβ€˜π·)π‘Ÿ)})
4826unissd 4918 . . . . . . . 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 3999 . . . . . . 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 23113 . . . . . . 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 3964 . . . . . . . . . 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 4609 . . . . . . . . 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 4059 . . . . . . . . 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 36940 . . 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 3947   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908  dom cdm 5676  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  Fincfn 8941  1c1 11113  β„*cxr 11251  β„•cn 12216  β„+crp 12978  βˆžMetcxmet 21129  Metcmet 21130  ballcbl 21131  MetOpencmopn 21134  β‡π‘‘clm 22950  Compccmp 23110  Cauccau 24994  CMetccmet 24995  TotBndctotbnd 36937
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cc 10432  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-acn 9939  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ico 13334  df-fz 13489  df-fl 13761  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-rlim 15437  df-rest 17372  df-topgen 17393  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-lm 22953  df-cmp 23111  df-fil 23570  df-fm 23662  df-flim 23663  df-flf 23664  df-cfil 24996  df-cau 24997  df-cmet 24998  df-totbnd 36939
This theorem is referenced by:  heibor  36992
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