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Theorem sstotbnd 37782
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 37781 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
3 elfpw 9394 . . . . . . . . 9 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑢𝑋𝑢 ∈ Fin))
43simprbi 496 . . . . . . . 8 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ∈ Fin)
5 mptfi 9391 . . . . . . . 8 (𝑢 ∈ Fin → (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
6 rnfi 9380 . . . . . . . 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 728 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
9 simprr 773 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
10 eqid 2737 . . . . . . . 8 (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
1110rnmpt 5968 . . . . . . 7 ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)}
123simplbi 497 . . . . . . . . . 10 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢𝑋)
13 ssrexv 4053 . . . . . . . . . 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 728 . . . . . . . 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 4066 . . . . . . 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 4022 . . . . . 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 7464 . . . . . . . . . . 11 (𝑥(ball‘𝑀)𝑑) ∈ V
2019dfiun3 5980 . . . . . . . . . 10 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
2118, 20eqtr4di 2795 . . . . . . . . 9 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑣 = 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
2221sseq2d 4016 . . . . . . . 8 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑌 𝑣𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
23 ssabral 4065 . . . . . . . . 9 (𝑣 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
24 sseq1 4009 . . . . . . . . 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 3622 . . . . . 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 837 . . . . 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 3156 . . . 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 7438 . . . . . . . . . 10 (𝑥 = (𝑓𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓𝑏)(ball‘𝑀)𝑑))
3130eqeq2d 2748 . . . . . . . . 9 (𝑥 = (𝑓𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3231ac6sfi 9320 . . . . . . . 8 ((𝑣 ∈ Fin ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3332adantrl 716 . . . . . . 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 6743 . . . . . . . . 9 (𝑓:𝑣𝑋 → ran 𝑓𝑋)
3635ad2antrl 728 . . . . . . . 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 777 . . . . . . . . 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 6736 . . . . . . . . . . 11 (𝑓:𝑣𝑋𝑓 Fn 𝑣)
3938ad2antrl 728 . . . . . . . . . 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 6826 . . . . . . . . . 10 (𝑓 Fn 𝑣𝑓:𝑣onto→ran 𝑓)
4139, 40sylib 218 . . . . . . . . 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 9351 . . . . . . . . 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 9394 . . . . . . . 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 781 . . . . . . . . . 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 5058 . . . . . . . . . . 11 𝑣 = 𝑏𝑣 𝑏
49 iuneq2 5011 . . . . . . . . . . 11 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑏𝑣 𝑏 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5048, 49eqtrid 2789 . . . . . . . . . 10 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑣 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5150ad2antll 729 . . . . . . . . 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 4020 . . . . . . . 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 2827 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑏) → (𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
5453rexrn 7107 . . . . . . . . . . 11 (𝑓 Fn 𝑣 → (∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏𝑣 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
55 eliun 4995 . . . . . . . . . . 11 (𝑦 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑))
56 eliun 4995 . . . . . . . . . . 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 2735 . . . . . . . . 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 4021 . . . . . . 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 5008 . . . . . . . . 9 (𝑢 = ran 𝑓 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
6261sseq2d 4016 . . . . . . . 8 (𝑢 = ran 𝑓 → (𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)))
6362rspcev 3622 . . . . . . 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 1935 . . . . 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 3156 . . . 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 212 . . 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 206  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   ciun 4991  cmpt 5225   × cxp 5683  ran crn 5686  cres 5687   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  Fincfn 8985  +crp 13034  Metcmet 21350  ballcbl 21351  TotBndctotbnd 37773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-totbnd 37775
This theorem is referenced by:  totbndss  37784
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