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Theorem sstotbnd 36234
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 36233 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
3 elfpw 9298 . . . . . . . . 9 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑢𝑋𝑢 ∈ Fin))
43simprbi 497 . . . . . . . 8 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ∈ Fin)
5 mptfi 9295 . . . . . . . 8 (𝑢 ∈ Fin → (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
6 rnfi 9279 . . . . . . . 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 726 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
9 simprr 771 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
10 eqid 2736 . . . . . . . 8 (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
1110rnmpt 5910 . . . . . . 7 ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)}
123simplbi 498 . . . . . . . . . 10 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢𝑋)
13 ssrexv 4011 . . . . . . . . . 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 726 . . . . . . . 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 4020 . . . . . . 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 3992 . . . . . 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 4876 . . . . . . . . . 10 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)))
19 ovex 7390 . . . . . . . . . . 11 (𝑥(ball‘𝑀)𝑑) ∈ V
2019dfiun3 5921 . . . . . . . . . 10 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
2118, 20eqtr4di 2794 . . . . . . . . 9 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑣 = 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
2221sseq2d 3976 . . . . . . . 8 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑌 𝑣𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
23 ssabral 4019 . . . . . . . . 9 (𝑣 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
24 sseq1 3969 . . . . . . . . 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 284 . . . . . . . 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 631 . . . . . . 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 3581 . . . . . 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 835 . . . . 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 3153 . . . 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 7364 . . . . . . . . . 10 (𝑥 = (𝑓𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓𝑏)(ball‘𝑀)𝑑))
3130eqeq2d 2747 . . . . . . . . 9 (𝑥 = (𝑓𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3231ac6sfi 9231 . . . . . . . 8 ((𝑣 ∈ Fin ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3332adantrl 714 . . . . . . 7 ((𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → ∃𝑓(𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3433adantl 482 . . . . . 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 6675 . . . . . . . . 9 (𝑓:𝑣𝑋 → ran 𝑓𝑋)
3635ad2antrl 726 . . . . . . . 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 775 . . . . . . . . 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 6668 . . . . . . . . . . 11 (𝑓:𝑣𝑋𝑓 Fn 𝑣)
3938ad2antrl 726 . . . . . . . . . 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 6762 . . . . . . . . . 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 9282 . . . . . . . . 9 ((𝑣 ∈ Fin ∧ 𝑓:𝑣onto→ran 𝑓) → ran 𝑓 ∈ Fin)
4337, 41, 42syl2anc 584 . . . . . . . 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 9298 . . . . . . . 8 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓𝑋 ∧ ran 𝑓 ∈ Fin))
4536, 43, 44sylanbrc 583 . . . . . . 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 779 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → 𝑌 𝑣)
4746adantr 481 . . . . . . . . 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 5018 . . . . . . . . . . 11 𝑣 = 𝑏𝑣 𝑏
49 iuneq2 4973 . . . . . . . . . . 11 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑏𝑣 𝑏 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5048, 49eqtrid 2788 . . . . . . . . . 10 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑣 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5150ad2antll 727 . . . . . . . . 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 3984 . . . . . . . 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 2823 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑏) → (𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
5453rexrn 7037 . . . . . . . . . . 11 (𝑓 Fn 𝑣 → (∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏𝑣 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
55 eliun 4958 . . . . . . . . . . 11 (𝑦 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑))
56 eliun 4958 . . . . . . . . . . 11 (𝑦 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏𝑣 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑))
5754, 55, 563bitr4g 313 . . . . . . . . . 10 (𝑓 Fn 𝑣 → (𝑦 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑦 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑)))
5857eqrdv 2734 . . . . . . . . 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 3985 . . . . . . 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 4970 . . . . . . . . 9 (𝑢 = ran 𝑓 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
6261sseq2d 3976 . . . . . . . 8 (𝑢 = ran 𝑓 → (𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)))
6362rspcev 3581 . . . . . . 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 584 . . . . . 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 1938 . . . . 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 3153 . . . 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 3174 . 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 278 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 396   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  wrex 3073  cin 3909  wss 3910  𝒫 cpw 4560   cuni 4865   ciun 4954  cmpt 5188   × cxp 5631  ran crn 5634  cres 5635   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  Fincfn 8883  +crp 12915  Metcmet 20782  ballcbl 20783  TotBndctotbnd 36225
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-2 12216  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-totbnd 36227
This theorem is referenced by:  totbndss  36236
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