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Theorem sstotbnd 36263
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 36262 . 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 9305 . . . . . . . . 9 (𝑒 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑒 βŠ† 𝑋 ∧ 𝑒 ∈ Fin))
43simprbi 498 . . . . . . . 8 (𝑒 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑒 ∈ Fin)
5 mptfi 9302 . . . . . . . 8 (𝑒 ∈ Fin β†’ (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) ∈ Fin)
6 rnfi 9286 . . . . . . . 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 727 . . . . . 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 772 . . . . . 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 2737 . . . . . . . 8 (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑))
1110rnmpt 5915 . . . . . . 7 ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) = {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑒 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)}
123simplbi 499 . . . . . . . . . 10 (𝑒 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑒 βŠ† 𝑋)
13 ssrexv 4016 . . . . . . . . . 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 727 . . . . . . . 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 4025 . . . . . . 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 3997 . . . . . 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 4881 . . . . . . . . . 10 (𝑣 = ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆͺ 𝑣 = βˆͺ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)))
19 ovex 7395 . . . . . . . . . . 11 (π‘₯(ballβ€˜π‘€)𝑑) ∈ V
2019dfiun3 5926 . . . . . . . . . 10 βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑))
2118, 20eqtr4di 2795 . . . . . . . . 9 (𝑣 = ran (π‘₯ ∈ 𝑒 ↦ (π‘₯(ballβ€˜π‘€)𝑑)) β†’ βˆͺ 𝑣 = βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑))
2221sseq2d 3981 . . . . . . . 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 4024 . . . . . . . . 9 (𝑣 βŠ† {𝑏 ∣ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)} ↔ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))
24 sseq1 3974 . . . . . . . . 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 285 . . . . . . . 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 632 . . . . . . 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 3584 . . . . . 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 836 . . . . 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 3154 . . . 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 7369 . . . . . . . . . 10 (π‘₯ = (π‘“β€˜π‘) β†’ (π‘₯(ballβ€˜π‘€)𝑑) = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
3130eqeq2d 2748 . . . . . . . . 9 (π‘₯ = (π‘“β€˜π‘) β†’ (𝑏 = (π‘₯(ballβ€˜π‘€)𝑑) ↔ 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
3231ac6sfi 9238 . . . . . . . 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 715 . . . . . . 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 483 . . . . . 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 6680 . . . . . . . . 9 (𝑓:π‘£βŸΆπ‘‹ β†’ ran 𝑓 βŠ† 𝑋)
3635ad2antrl 727 . . . . . . . 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 776 . . . . . . . . 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 6673 . . . . . . . . . . 11 (𝑓:π‘£βŸΆπ‘‹ β†’ 𝑓 Fn 𝑣)
3938ad2antrl 727 . . . . . . . . . 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 6767 . . . . . . . . . 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 9289 . . . . . . . . 9 ((𝑣 ∈ Fin ∧ 𝑓:𝑣–ontoβ†’ran 𝑓) β†’ ran 𝑓 ∈ Fin)
4337, 41, 42syl2anc 585 . . . . . . . 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 9305 . . . . . . . 8 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓 βŠ† 𝑋 ∧ ran 𝑓 ∈ Fin))
4536, 43, 44sylanbrc 584 . . . . . . 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 780 . . . . . . . . . 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 482 . . . . . . . . 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 5023 . . . . . . . . . . 11 βˆͺ 𝑣 = βˆͺ 𝑏 ∈ 𝑣 𝑏
49 iuneq2 4978 . . . . . . . . . . 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 2789 . . . . . . . . . 10 (βˆ€π‘ ∈ 𝑣 𝑏 = ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑) β†’ βˆͺ 𝑣 = βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
5150ad2antll 728 . . . . . . . . 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 3989 . . . . . . . 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 2824 . . . . . . . . . . . 12 (π‘₯ = (π‘“β€˜π‘) β†’ (𝑦 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ↔ 𝑦 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
5453rexrn 7042 . . . . . . . . . . 11 (𝑓 Fn 𝑣 β†’ (βˆƒπ‘₯ ∈ ran 𝑓 𝑦 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘ ∈ 𝑣 𝑦 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑)))
55 eliun 4963 . . . . . . . . . . 11 (𝑦 ∈ βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘₯ ∈ ran 𝑓 𝑦 ∈ (π‘₯(ballβ€˜π‘€)𝑑))
56 eliun 4963 . . . . . . . . . . 11 (𝑦 ∈ βˆͺ 𝑏 ∈ 𝑣 ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘ ∈ 𝑣 𝑦 ∈ ((π‘“β€˜π‘)(ballβ€˜π‘€)𝑑))
5754, 55, 563bitr4g 314 . . . . . . . . . 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 2735 . . . . . . . . 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 3990 . . . . . . 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 4975 . . . . . . . . 9 (𝑒 = ran 𝑓 β†’ βˆͺ π‘₯ ∈ 𝑒 (π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ π‘₯ ∈ ran 𝑓(π‘₯(ballβ€˜π‘€)𝑑))
6261sseq2d 3981 . . . . . . . 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 3584 . . . . . . 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 585 . . . . . 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 1939 . . . . 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 3154 . . . 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 3175 . 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 279 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  βˆƒwrex 3074   ∩ cin 3914   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870  βˆͺ ciun 4959   ↦ cmpt 5193   Γ— cxp 5636  ran crn 5639   β†Ύ cres 5640   Fn wfn 6496  βŸΆwf 6497  β€“ontoβ†’wfo 6499  β€˜cfv 6501  (class class class)co 7362  Fincfn 8890  β„+crp 12922  Metcmet 20798  ballcbl 20799  TotBndctotbnd 36254
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-2 12223  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-totbnd 36256
This theorem is referenced by:  totbndss  36265
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