Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sstotbnd Structured version   Visualization version   GIF version

Theorem sstotbnd 36581
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 ↾ (𝑌 × 𝑌))
Assertion
Ref Expression
sstotbnd ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
Distinct variable groups:   𝑏,𝑑,𝑣,𝑥,𝑀   𝑋,𝑏,𝑑,𝑣,𝑥   𝑁,𝑑,𝑣,𝑥   𝑌,𝑏,𝑑,𝑣,𝑥
Allowed substitution hint:   𝑁(𝑏)

Proof of Theorem sstotbnd
Dummy variables 𝑓 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstotbnd.2 . . 3 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
21sstotbnd2 36580 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
3 elfpw 9350 . . . . . . . . 9 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑢𝑋𝑢 ∈ Fin))
43simprbi 498 . . . . . . . 8 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ∈ Fin)
5 mptfi 9347 . . . . . . . 8 (𝑢 ∈ Fin → (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
6 rnfi 9331 . . . . . . . 8 ((𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin → ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
74, 5, 63syl 18 . . . . . . 7 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
87ad2antrl 727 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
9 simprr 772 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
10 eqid 2733 . . . . . . . 8 (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
1110rnmpt 5952 . . . . . . 7 ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)}
123simplbi 499 . . . . . . . . . 10 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢𝑋)
13 ssrexv 4050 . . . . . . . . . 10 (𝑢𝑋 → (∃𝑥𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
1412, 13syl 17 . . . . . . . . 9 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → (∃𝑥𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
1514ad2antrl 727 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → (∃𝑥𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
1615ss2abdv 4059 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → {𝑏 ∣ ∃𝑥𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)} ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})
1711, 16eqsstrid 4029 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})
18 unieq 4918 . . . . . . . . . 10 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)))
19 ovex 7437 . . . . . . . . . . 11 (𝑥(ball‘𝑀)𝑑) ∈ V
2019dfiun3 5963 . . . . . . . . . 10 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
2118, 20eqtr4di 2791 . . . . . . . . 9 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑣 = 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
2221sseq2d 4013 . . . . . . . 8 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑌 𝑣𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
23 ssabral 4058 . . . . . . . . 9 (𝑣 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
24 sseq1 4006 . . . . . . . . 9 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑣 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}))
2523, 24bitr3id 285 . . . . . . . 8 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}))
2622, 25anbi12d 632 . . . . . . 7 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ((𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ (𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) ∧ ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})))
2726rspcev 3612 . . . . . 6 ((ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin ∧ (𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) ∧ ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})) → ∃𝑣 ∈ Fin (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
288, 9, 17, 27syl12anc 836 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → ∃𝑣 ∈ Fin (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
2928rexlimdvaa 3157 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ Fin (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
30 oveq1 7411 . . . . . . . . . 10 (𝑥 = (𝑓𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓𝑏)(ball‘𝑀)𝑑))
3130eqeq2d 2744 . . . . . . . . 9 (𝑥 = (𝑓𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3231ac6sfi 9283 . . . . . . . 8 ((𝑣 ∈ Fin ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3332adantrl 715 . . . . . . 7 ((𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → ∃𝑓(𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3433adantl 483 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → ∃𝑓(𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
35 frn 6721 . . . . . . . . 9 (𝑓:𝑣𝑋 → ran 𝑓𝑋)
3635ad2antrl 727 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → ran 𝑓𝑋)
37 simplrl 776 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → 𝑣 ∈ Fin)
38 ffn 6714 . . . . . . . . . . 11 (𝑓:𝑣𝑋𝑓 Fn 𝑣)
3938ad2antrl 727 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → 𝑓 Fn 𝑣)
40 dffn4 6808 . . . . . . . . . 10 (𝑓 Fn 𝑣𝑓:𝑣onto→ran 𝑓)
4139, 40sylib 217 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → 𝑓:𝑣onto→ran 𝑓)
42 fofi 9334 . . . . . . . . 9 ((𝑣 ∈ Fin ∧ 𝑓:𝑣onto→ran 𝑓) → ran 𝑓 ∈ Fin)
4337, 41, 42syl2anc 585 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → ran 𝑓 ∈ Fin)
44 elfpw 9350 . . . . . . . 8 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓𝑋 ∧ ran 𝑓 ∈ Fin))
4536, 43, 44sylanbrc 584 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin))
46 simprrl 780 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → 𝑌 𝑣)
4746adantr 482 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → 𝑌 𝑣)
48 uniiun 5060 . . . . . . . . . . 11 𝑣 = 𝑏𝑣 𝑏
49 iuneq2 5015 . . . . . . . . . . 11 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑏𝑣 𝑏 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5048, 49eqtrid 2785 . . . . . . . . . 10 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑣 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5150ad2antll 728 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → 𝑣 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5247, 51sseqtrd 4021 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → 𝑌 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5330eleq2d 2820 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑏) → (𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
5453rexrn 7084 . . . . . . . . . . 11 (𝑓 Fn 𝑣 → (∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏𝑣 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
55 eliun 5000 . . . . . . . . . . 11 (𝑦 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑))
56 eliun 5000 . . . . . . . . . . 11 (𝑦 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏𝑣 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑))
5754, 55, 563bitr4g 314 . . . . . . . . . 10 (𝑓 Fn 𝑣 → (𝑦 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑦 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑)))
5857eqrdv 2731 . . . . . . . . 9 (𝑓 Fn 𝑣 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5939, 58syl 17 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
6052, 59sseqtrrd 4022 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → 𝑌 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
61 iuneq1 5012 . . . . . . . . 9 (𝑢 = ran 𝑓 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
6261sseq2d 4013 . . . . . . . 8 (𝑢 = ran 𝑓 → (𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)))
6362rspcev 3612 . . . . . . 7 ((ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
6445, 60, 63syl2anc 585 . . . . . 6 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))) → ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
6534, 64exlimddv 1939 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
6665rexlimdvaa 3157 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑣 ∈ Fin (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
6729, 66impbid 211 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑣 ∈ Fin (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
6867ralbidv 3178 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∀𝑑 ∈ ℝ+𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
692, 68bitrd 279 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3062  wrex 3071  cin 3946  wss 3947  𝒫 cpw 4601   cuni 4907   ciun 4996  cmpt 5230   × cxp 5673  ran crn 5676  cres 5677   Fn wfn 6535  wf 6536  ontowfo 6538  cfv 6540  (class class class)co 7404  Fincfn 8935  +crp 12970  Metcmet 20915  ballcbl 20916  TotBndctotbnd 36572
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 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  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
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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-iun 4998  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-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-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-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  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-2 12271  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-psmet 20921  df-xmet 20922  df-met 20923  df-bl 20924  df-totbnd 36574
This theorem is referenced by:  totbndss  36583
  Copyright terms: Public domain W3C validator