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Theorem sstotbnd 32744
 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 32743 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
3 elfpw 8151 . . . . . . . . 9 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑢𝑋𝑢 ∈ Fin))
43simprbi 479 . . . . . . . 8 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ∈ Fin)
5 mptfi 8148 . . . . . . . 8 (𝑢 ∈ Fin → (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
6 rnfi 8132 . . . . . . . 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 760 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
9 simprr 792 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
10 eqid 2610 . . . . . . . 8 (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
1110rnmpt 5292 . . . . . . 7 ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)}
123simplbi 475 . . . . . . . . . 10 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢𝑋)
13 ssrexv 3630 . . . . . . . . . 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 760 . . . . . . . 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 3638 . . . . . . 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, 16syl5eqss 3612 . . . . . 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 4380 . . . . . . . . . 10 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)))
19 ovex 6577 . . . . . . . . . . 11 (𝑥(ball‘𝑀)𝑑) ∈ V
2019dfiun3 5301 . . . . . . . . . 10 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
2118, 20syl6eqr 2662 . . . . . . . . 9 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑣 = 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
2221sseq2d 3596 . . . . . . . 8 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑌 𝑣𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
23 ssabral 3636 . . . . . . . . 9 (𝑣 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
24 sseq1 3589 . . . . . . . . 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, 24syl5bbr 273 . . . . . . . 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 743 . . . . . . 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 3282 . . . . . 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 1316 . . . . 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 3014 . . . 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 6556 . . . . . . . . . 10 (𝑥 = (𝑓𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓𝑏)(ball‘𝑀)𝑑))
3130eqeq2d 2620 . . . . . . . . 9 (𝑥 = (𝑓𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3231ac6sfi 8089 . . . . . . . 8 ((𝑣 ∈ Fin ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3332adantrl 748 . . . . . . 7 ((𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → ∃𝑓(𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3433adantl 481 . . . . . 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 5966 . . . . . . . . 9 (𝑓:𝑣𝑋 → ran 𝑓𝑋)
3635ad2antrl 760 . . . . . . . 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 796 . . . . . . . . 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 5958 . . . . . . . . . . 11 (𝑓:𝑣𝑋𝑓 Fn 𝑣)
3938ad2antrl 760 . . . . . . . . . 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 6034 . . . . . . . . . 10 (𝑓 Fn 𝑣𝑓:𝑣onto→ran 𝑓)
4139, 40sylib 207 . . . . . . . . 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 8135 . . . . . . . . 9 ((𝑣 ∈ Fin ∧ 𝑓:𝑣onto→ran 𝑓) → ran 𝑓 ∈ Fin)
4337, 41, 42syl2anc 691 . . . . . . . 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 8151 . . . . . . . 8 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓𝑋 ∧ ran 𝑓 ∈ Fin))
4536, 43, 44sylanbrc 695 . . . . . . 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 800 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → 𝑌 𝑣)
4746adantr 480 . . . . . . . . 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 4509 . . . . . . . . . . 11 𝑣 = 𝑏𝑣 𝑏
49 iuneq2 4473 . . . . . . . . . . 11 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑏𝑣 𝑏 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5048, 49syl5eq 2656 . . . . . . . . . 10 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑣 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5150ad2antll 761 . . . . . . . . 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 3604 . . . . . . . 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 2673 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑏) → (𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
5453rexrn 6269 . . . . . . . . . . 11 (𝑓 Fn 𝑣 → (∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏𝑣 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
55 eliun 4460 . . . . . . . . . . 11 (𝑦 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑))
56 eliun 4460 . . . . . . . . . . 11 (𝑦 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏𝑣 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑))
5754, 55, 563bitr4g 302 . . . . . . . . . 10 (𝑓 Fn 𝑣 → (𝑦 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑦 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑)))
5857eqrdv 2608 . . . . . . . . 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, 59sseqtr4d 3605 . . . . . . 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 4470 . . . . . . . . 9 (𝑢 = ran 𝑓 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
6261sseq2d 3596 . . . . . . . 8 (𝑢 = ran 𝑓 → (𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)))
6362rspcev 3282 . . . . . . 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 691 . . . . . 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 1850 . . . . 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 3014 . . . 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 201 . . 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 2969 . 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 267 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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  ran crn 5039   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℝ+crp 11708  Metcme 19553  ballcbl 19554  TotBndctotbnd 32735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-2 10956  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-totbnd 32737 This theorem is referenced by:  totbndss  32746
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