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Theorem sstotbnd 36729
Description: Condition for a subset of a metric space to be totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Hypothesis
Ref Expression
sstotbnd.2 𝑁 = (𝑀 β†Ύ (π‘Œ Γ— π‘Œ))
Assertion
Ref Expression
sstotbnd ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑁 ∈ (TotBndβ€˜π‘Œ) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
Distinct variable groups:   𝑏,𝑑,𝑣,π‘₯,𝑀   𝑋,𝑏,𝑑,𝑣,π‘₯   𝑁,𝑑,𝑣,π‘₯   π‘Œ,𝑏,𝑑,𝑣,π‘₯
Allowed substitution hint:   𝑁(𝑏)

Proof of Theorem sstotbnd
Dummy variables 𝑓 𝑒 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstotbnd.2 . . 3 𝑁 = (𝑀 β†Ύ (π‘Œ Γ— π‘Œ))
21sstotbnd2 36728 . 2 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑁 ∈ (TotBndβ€˜π‘Œ) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘’ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑)))
3 elfpw 9356 . . . . . . . . 9 (𝑒 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑒 βŠ† 𝑋 ∧ 𝑒 ∈ Fin))
43simprbi 497 . . . . . . . 8 (𝑒 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑒 ∈ Fin)
5 mptfi 9353 . . . . . . . 8 (𝑒 ∈ Fin β†’ (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin)
6 rnfi 9337 . . . . . . . 8 ((π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin β†’ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin)
74, 5, 63syl 18 . . . . . . 7 (𝑒 ∈ (𝒫 𝑋 ∩ Fin) β†’ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin)
87ad2antrl 726 . . . . . 6 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑒 ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))) β†’ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin)
9 simprr 771 . . . . . 6 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑒 ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))) β†’ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))
10 eqid 2732 . . . . . . . 8 (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑))
1110rnmpt 5954 . . . . . . 7 ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑒 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)}
123simplbi 498 . . . . . . . . . 10 (𝑒 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑒 βŠ† 𝑋)
13 ssrexv 4051 . . . . . . . . . 10 (𝑒 βŠ† 𝑋 β†’ (βˆƒπ‘₯ ∈ 𝑒 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
1412, 13syl 17 . . . . . . . . 9 (𝑒 ∈ (𝒫 𝑋 ∩ Fin) β†’ (βˆƒπ‘₯ ∈ 𝑒 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
1514ad2antrl 726 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑒 ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))) β†’ (βˆƒπ‘₯ ∈ 𝑒 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
1615ss2abdv 4060 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑒 ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))) β†’ {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑒 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)} βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)})
1711, 16eqsstrid 4030 . . . . . 6 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑒 ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))) β†’ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)})
18 unieq 4919 . . . . . . . . . 10 (𝑣 = ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆͺ 𝑣 = βˆͺ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)))
19 ovex 7444 . . . . . . . . . . 11 (π‘₯(ballβ€˜π‘€)𝑑) ∈ V
2019dfiun3 5965 . . . . . . . . . 10 βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑))
2118, 20eqtr4di 2790 . . . . . . . . 9 (𝑣 = ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆͺ 𝑣 = βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))
2221sseq2d 4014 . . . . . . . 8 (𝑣 = ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ (π‘Œ βŠ† βˆͺ 𝑣 ↔ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑)))
23 ssabral 4059 . . . . . . . . 9 (𝑣 βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)} ↔ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))
24 sseq1 4007 . . . . . . . . 9 (𝑣 = ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ (𝑣 βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)} ↔ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)}))
2523, 24bitr3id 284 . . . . . . . 8 (𝑣 = ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ (βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) ↔ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)}))
2622, 25anbi12d 631 . . . . . . 7 (𝑣 = ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ ((π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)) ↔ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) ∧ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)})))
2726rspcev 3612 . . . . . 6 ((ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin ∧ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) ∧ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)})) β†’ βˆƒπ‘£ ∈ Fin (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
288, 9, 17, 27syl12anc 835 . . . . 5 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑒 ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))) β†’ βˆƒπ‘£ ∈ Fin (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
2928rexlimdvaa 3156 . . . 4 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆƒπ‘’ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘£ ∈ Fin (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
30 oveq1 7418 . . . . . . . . . 10 (π‘₯ = (π‘“β€˜π‘) β†’ (π‘₯(ballβ€˜π‘€)𝑑) = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
3130eqeq2d 2743 . . . . . . . . 9 (π‘₯ = (π‘“β€˜π‘) β†’ (𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) ↔ 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
3231ac6sfi 9289 . . . . . . . 8 ((𝑣 ∈ Fin ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆƒπ‘“(𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
3332adantrl 714 . . . . . . 7 ((𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))) β†’ βˆƒπ‘“(𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
3433adantl 482 . . . . . 6 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) β†’ βˆƒπ‘“(𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
35 frn 6724 . . . . . . . . 9 (𝑓:π‘£βŸΆπ‘‹ β†’ ran 𝑓 βŠ† 𝑋)
3635ad2antrl 726 . . . . . . . 8 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ ran 𝑓 βŠ† 𝑋)
37 simplrl 775 . . . . . . . . 9 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ 𝑣 ∈ Fin)
38 ffn 6717 . . . . . . . . . . 11 (𝑓:π‘£βŸΆπ‘‹ β†’ 𝑓 Fn 𝑣)
3938ad2antrl 726 . . . . . . . . . 10 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ 𝑓 Fn 𝑣)
40 dffn4 6811 . . . . . . . . . 10 (𝑓 Fn 𝑣 ↔ 𝑓:𝑣–ontoβ†’ran 𝑓)
4139, 40sylib 217 . . . . . . . . 9 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ 𝑓:𝑣–ontoβ†’ran 𝑓)
42 fofi 9340 . . . . . . . . 9 ((𝑣 ∈ Fin ∧ 𝑓:𝑣–ontoβ†’ran 𝑓) β†’ ran 𝑓 ∈ Fin)
4337, 41, 42syl2anc 584 . . . . . . . 8 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ ran 𝑓 ∈ Fin)
44 elfpw 9356 . . . . . . . 8 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓 βŠ† 𝑋 ∧ ran 𝑓 ∈ Fin))
4536, 43, 44sylanbrc 583 . . . . . . 7 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin))
46 simprrl 779 . . . . . . . . . 10 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) β†’ π‘Œ βŠ† βˆͺ 𝑣)
4746adantr 481 . . . . . . . . 9 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ π‘Œ βŠ† βˆͺ 𝑣)
48 uniiun 5061 . . . . . . . . . . 11 βˆͺ 𝑣 = βˆͺ 𝑏 ∈ 𝑣 𝑏
49 iuneq2 5016 . . . . . . . . . . 11 (βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑) β†’ βˆͺ 𝑏 ∈ 𝑣 𝑏 = βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
5048, 49eqtrid 2784 . . . . . . . . . 10 (βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑) β†’ βˆͺ 𝑣 = βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
5150ad2antll 727 . . . . . . . . 9 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ βˆͺ 𝑣 = βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
5247, 51sseqtrd 4022 . . . . . . . 8 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ π‘Œ βŠ† βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
5330eleq2d 2819 . . . . . . . . . . . 12 (π‘₯ = (π‘“β€˜π‘) β†’ (𝑦 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ↔ 𝑦 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
5453rexrn 7088 . . . . . . . . . . 11 (𝑓 Fn 𝑣 β†’ (βˆƒπ‘₯ ∈ ran 𝑓 𝑦 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘ ∈ 𝑣 𝑦 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
55 eliun 5001 . . . . . . . . . . 11 (𝑦 ∈ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘₯ ∈ ran 𝑓 𝑦 ∈ (π‘₯(ballβ€˜π‘€)𝑑))
56 eliun 5001 . . . . . . . . . . 11 (𝑦 ∈ βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘ ∈ 𝑣 𝑦 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
5754, 55, 563bitr4g 313 . . . . . . . . . 10 (𝑓 Fn 𝑣 β†’ (𝑦 ∈ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) ↔ 𝑦 ∈ βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
5857eqrdv 2730 . . . . . . . . 9 (𝑓 Fn 𝑣 β†’ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
5939, 58syl 17 . . . . . . . 8 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
6052, 59sseqtrrd 4023 . . . . . . 7 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ π‘Œ βŠ† βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑))
61 iuneq1 5013 . . . . . . . . 9 (𝑒 = ran 𝑓 β†’ βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑))
6261sseq2d 4014 . . . . . . . 8 (𝑒 = ran 𝑓 β†’ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) ↔ π‘Œ βŠ† βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑)))
6362rspcev 3612 . . . . . . 7 ((ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆƒπ‘’ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))
6445, 60, 63syl2anc 584 . . . . . 6 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) ∧ (𝑓:π‘£βŸΆπ‘‹ ∧ βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))) β†’ βˆƒπ‘’ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))
6534, 64exlimddv 1938 . . . . 5 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (𝑣 ∈ Fin ∧ (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))) β†’ βˆƒπ‘’ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))
6665rexlimdvaa 3156 . . . 4 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆƒπ‘£ ∈ Fin (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆƒπ‘’ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑)))
6729, 66impbid 211 . . 3 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆƒπ‘’ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘£ ∈ Fin (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
6867ralbidv 3177 . 2 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘’ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
692, 68bitrd 278 1 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑁 ∈ (TotBndβ€˜π‘Œ) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (π‘Œ βŠ† βˆͺ 𝑣 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908  βˆͺ ciun 4997   ↦ cmpt 5231   Γ— cxp 5674  ran crn 5677   β†Ύ cres 5678   Fn wfn 6538  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7411  Fincfn 8941  β„+crp 12976  Metcmet 20936  ballcbl 20937  TotBndctotbnd 36720
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-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  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
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-iun 4999  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-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-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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-2 12277  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-totbnd 36722
This theorem is referenced by:  totbndss  36731
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