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Theorem istotbnd3 36276
Description: A metric space is totally bounded iff there is a finite Ξ΅-net for every positive Ξ΅. This differs from the definition in providing a finite set of ball centers rather than a finite set of balls. (Contributed by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
istotbnd3 (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
Distinct variable groups:   𝑣,𝑑,π‘₯,𝑀   𝑋,𝑑,𝑣,π‘₯

Proof of Theorem istotbnd3
Dummy variables 𝑏 𝑓 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istotbnd 36274 . 2 (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘€ ∈ Fin (βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
2 oveq1 7365 . . . . . . . . . . . 12 (π‘₯ = (π‘“β€˜π‘) β†’ (π‘₯(ballβ€˜π‘€)𝑑) = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
32eqeq2d 2744 . . . . . . . . . . 11 (π‘₯ = (π‘“β€˜π‘) β†’ (𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) ↔ 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
43ac6sfi 9234 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆƒπ‘“(𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
54ex 414 . . . . . . . . 9 (𝑀 ∈ Fin β†’ (βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘“(𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))))
65ad2antlr 726 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ βˆͺ 𝑀 = 𝑋) β†’ (βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘“(𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))))
7 simprrl 780 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ 𝑓:π‘€βŸΆπ‘‹)
87frnd 6677 . . . . . . . . . . . 12 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ ran 𝑓 βŠ† 𝑋)
9 simplr 768 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ 𝑀 ∈ Fin)
107ffnd 6670 . . . . . . . . . . . . . 14 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ 𝑓 Fn 𝑀)
11 dffn4 6763 . . . . . . . . . . . . . 14 (𝑓 Fn 𝑀 ↔ 𝑓:𝑀–ontoβ†’ran 𝑓)
1210, 11sylib 217 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ 𝑓:𝑀–ontoβ†’ran 𝑓)
13 fofi 9285 . . . . . . . . . . . . 13 ((𝑀 ∈ Fin ∧ 𝑓:𝑀–ontoβ†’ran 𝑓) β†’ ran 𝑓 ∈ Fin)
149, 12, 13syl2anc 585 . . . . . . . . . . . 12 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ ran 𝑓 ∈ Fin)
15 elfpw 9301 . . . . . . . . . . . 12 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓 βŠ† 𝑋 ∧ ran 𝑓 ∈ Fin))
168, 14, 15sylanbrc 584 . . . . . . . . . . 11 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin))
172eleq2d 2820 . . . . . . . . . . . . . . . 16 (π‘₯ = (π‘“β€˜π‘) β†’ (𝑣 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ↔ 𝑣 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
1817rexrn 7038 . . . . . . . . . . . . . . 15 (𝑓 Fn 𝑀 β†’ (βˆƒπ‘₯ ∈ ran 𝑓 𝑣 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘ ∈ 𝑀 𝑣 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
1910, 18syl 17 . . . . . . . . . . . . . 14 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ (βˆƒπ‘₯ ∈ ran 𝑓 𝑣 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘ ∈ 𝑀 𝑣 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
20 eliun 4959 . . . . . . . . . . . . . 14 (𝑣 ∈ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘₯ ∈ ran 𝑓 𝑣 ∈ (π‘₯(ballβ€˜π‘€)𝑑))
21 eliun 4959 . . . . . . . . . . . . . 14 (𝑣 ∈ βˆͺ 𝑏 ∈ 𝑀 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘ ∈ 𝑀 𝑣 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
2219, 20, 213bitr4g 314 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ (𝑣 ∈ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) ↔ 𝑣 ∈ βˆͺ 𝑏 ∈ 𝑀 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
2322eqrdv 2731 . . . . . . . . . . . 12 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ 𝑏 ∈ 𝑀 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
24 simprrr 781 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
25 iuneq2 4974 . . . . . . . . . . . . 13 (βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑) β†’ βˆͺ 𝑏 ∈ 𝑀 𝑏 = βˆͺ 𝑏 ∈ 𝑀 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
2624, 25syl 17 . . . . . . . . . . . 12 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ βˆͺ 𝑏 ∈ 𝑀 𝑏 = βˆͺ 𝑏 ∈ 𝑀 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
27 uniiun 5019 . . . . . . . . . . . . 13 βˆͺ 𝑀 = βˆͺ 𝑏 ∈ 𝑀 𝑏
28 simprl 770 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ βˆͺ 𝑀 = 𝑋)
2927, 28eqtr3id 2787 . . . . . . . . . . . 12 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ βˆͺ 𝑏 ∈ 𝑀 𝑏 = 𝑋)
3023, 26, 293eqtr2d 2779 . . . . . . . . . . 11 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)
31 iuneq1 4971 . . . . . . . . . . . . 13 (𝑣 = ran 𝑓 β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑))
3231eqeq1d 2735 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋 ↔ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
3332rspcev 3580 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) = 𝑋) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)
3416, 30, 33syl2anc 585 . . . . . . . . . 10 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ (βˆͺ 𝑀 = 𝑋 ∧ (𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)
3534expr 458 . . . . . . . . 9 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ βˆͺ 𝑀 = 𝑋) β†’ ((𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
3635exlimdv 1937 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ βˆͺ 𝑀 = 𝑋) β†’ (βˆƒπ‘“(𝑓:π‘€βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑀 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
376, 36syld 47 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) ∧ βˆͺ 𝑀 = 𝑋) β†’ (βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
3837expimpd 455 . . . . . 6 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑀 ∈ Fin) β†’ ((βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
3938rexlimdva 3149 . . . . 5 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (βˆƒπ‘€ ∈ Fin (βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
40 elfpw 9301 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 βŠ† 𝑋 ∧ 𝑣 ∈ Fin))
4140simprbi 498 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑣 ∈ Fin)
4241ad2antrl 727 . . . . . . . . 9 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)) β†’ 𝑣 ∈ Fin)
43 mptfi 9298 . . . . . . . . 9 (𝑣 ∈ Fin β†’ (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin)
44 rnfi 9282 . . . . . . . . 9 ((π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin β†’ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin)
4542, 43, 443syl 18 . . . . . . . 8 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)) β†’ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin)
46 ovex 7391 . . . . . . . . . 10 (π‘₯(ballβ€˜π‘€)𝑑) ∈ V
4746dfiun3 5922 . . . . . . . . 9 βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑))
48 simprr 772 . . . . . . . . 9 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)
4947, 48eqtr3id 2787 . . . . . . . 8 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)) β†’ βˆͺ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = 𝑋)
50 eqid 2733 . . . . . . . . . 10 (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑))
5150rnmpt 5911 . . . . . . . . 9 ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑣 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)}
5240simplbi 499 . . . . . . . . . . . 12 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑣 βŠ† 𝑋)
5352ad2antrl 727 . . . . . . . . . . 11 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)) β†’ 𝑣 βŠ† 𝑋)
54 ssrexv 4012 . . . . . . . . . . 11 (𝑣 βŠ† 𝑋 β†’ (βˆƒπ‘₯ ∈ 𝑣 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
5553, 54syl 17 . . . . . . . . . 10 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)) β†’ (βˆƒπ‘₯ ∈ 𝑣 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
5655ss2abdv 4021 . . . . . . . . 9 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)) β†’ {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑣 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)} βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)})
5751, 56eqsstrid 3993 . . . . . . . 8 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)) β†’ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)})
58 unieq 4877 . . . . . . . . . . 11 (𝑀 = ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆͺ 𝑀 = βˆͺ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)))
5958eqeq1d 2735 . . . . . . . . . 10 (𝑀 = ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ (βˆͺ 𝑀 = 𝑋 ↔ βˆͺ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = 𝑋))
60 ssabral 4020 . . . . . . . . . . 11 (𝑀 βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)} ↔ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))
61 sseq1 3970 . . . . . . . . . . 11 (𝑀 = ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ (𝑀 βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)} ↔ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)}))
6260, 61bitr3id 285 . . . . . . . . . 10 (𝑀 = ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ (βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) ↔ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)}))
6359, 62anbi12d 632 . . . . . . . . 9 (𝑀 = ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ ((βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)) ↔ (βˆͺ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = 𝑋 ∧ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)})))
6463rspcev 3580 . . . . . . . 8 ((ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin ∧ (βˆͺ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = 𝑋 ∧ ran (π‘₯ ∈ 𝑣 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)})) β†’ βˆƒπ‘€ ∈ Fin (βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
6545, 49, 57, 64syl12anc 836 . . . . . . 7 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)) β†’ βˆƒπ‘€ ∈ Fin (βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
6665expr 458 . . . . . 6 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋 β†’ βˆƒπ‘€ ∈ Fin (βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
6766rexlimdva 3149 . . . . 5 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋 β†’ βˆƒπ‘€ ∈ Fin (βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
6839, 67impbid 211 . . . 4 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (βˆƒπ‘€ ∈ Fin (βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)) ↔ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
6968ralbidv 3171 . . 3 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘€ ∈ Fin (βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
7069pm5.32i 576 . 2 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘€ ∈ Fin (βˆͺ 𝑀 = 𝑋 ∧ βˆ€π‘ ∈ 𝑀 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
711, 70bitri 275 1 (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866  βˆͺ ciun 4955   ↦ cmpt 5189  ran crn 5635   Fn wfn 6492  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358  Fincfn 8886  β„+crp 12920  Metcmet 20798  ballcbl 20799  TotBndctotbnd 36271
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-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-1st 7922  df-2nd 7923  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-fin 8890  df-totbnd 36273
This theorem is referenced by:  0totbnd  36278  sstotbnd2  36279  equivtotbnd  36283  totbndbnd  36294  prdstotbnd  36299
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