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Theorem sstotbnd 33206
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 33205 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
3 elfpw 8212 . . . . . . . . 9 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑢𝑋𝑢 ∈ Fin))
43simprbi 480 . . . . . . . 8 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ∈ Fin)
5 mptfi 8209 . . . . . . . 8 (𝑢 ∈ Fin → (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
6 rnfi 8193 . . . . . . . 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 763 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
9 simprr 795 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))) → 𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
10 eqid 2621 . . . . . . . 8 (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
1110rnmpt 5331 . . . . . . 7 ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)}
123simplbi 476 . . . . . . . . . 10 (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢𝑋)
13 ssrexv 3646 . . . . . . . . . 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 763 . . . . . . . 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 3654 . . . . . . 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 3628 . . . . . 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 4410 . . . . . . . . . 10 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)))
19 ovex 6632 . . . . . . . . . . 11 (𝑥(ball‘𝑀)𝑑) ∈ V
2019dfiun3 5340 . . . . . . . . . 10 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
2118, 20syl6eqr 2673 . . . . . . . . 9 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑣 = 𝑥𝑢 (𝑥(ball‘𝑀)𝑑))
2221sseq2d 3612 . . . . . . . 8 (𝑣 = ran (𝑥𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑌 𝑣𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑)))
23 ssabral 3652 . . . . . . . . 9 (𝑣 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
24 sseq1 3605 . . . . . . . . 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 274 . . . . . . . 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 746 . . . . . . 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 3295 . . . . . 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 1321 . . . . 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 3025 . . . 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 6611 . . . . . . . . . 10 (𝑥 = (𝑓𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓𝑏)(ball‘𝑀)𝑑))
3130eqeq2d 2631 . . . . . . . . 9 (𝑥 = (𝑓𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3231ac6sfi 8148 . . . . . . . 8 ((𝑣 ∈ Fin ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑣𝑋 ∧ ∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
3332adantrl 751 . . . . . . 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 6010 . . . . . . . . 9 (𝑓:𝑣𝑋 → ran 𝑓𝑋)
3635ad2antrl 763 . . . . . . . 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 799 . . . . . . . . 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 6002 . . . . . . . . . . 11 (𝑓:𝑣𝑋𝑓 Fn 𝑣)
3938ad2antrl 763 . . . . . . . . . 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 6078 . . . . . . . . . 10 (𝑓 Fn 𝑣𝑓:𝑣onto→ran 𝑓)
4139, 40sylib 208 . . . . . . . . 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 8196 . . . . . . . . 9 ((𝑣 ∈ Fin ∧ 𝑓:𝑣onto→ran 𝑓) → ran 𝑓 ∈ Fin)
4337, 41, 42syl2anc 692 . . . . . . . 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 8212 . . . . . . . 8 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓𝑋 ∧ ran 𝑓 ∈ Fin))
4536, 43, 44sylanbrc 697 . . . . . . 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 803 . . . . . . . . . 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 4539 . . . . . . . . . . 11 𝑣 = 𝑏𝑣 𝑏
49 iuneq2 4503 . . . . . . . . . . 11 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑏𝑣 𝑏 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5048, 49syl5eq 2667 . . . . . . . . . 10 (∀𝑏𝑣 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑣 = 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑))
5150ad2antll 764 . . . . . . . . 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 3620 . . . . . . . 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 2684 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑏) → (𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
5453rexrn 6317 . . . . . . . . . . 11 (𝑓 Fn 𝑣 → (∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏𝑣 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
55 eliun 4490 . . . . . . . . . . 11 (𝑦 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑))
56 eliun 4490 . . . . . . . . . . 11 (𝑦 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏𝑣 𝑦 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑))
5754, 55, 563bitr4g 303 . . . . . . . . . 10 (𝑓 Fn 𝑣 → (𝑦 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑦 𝑏𝑣 ((𝑓𝑏)(ball‘𝑀)𝑑)))
5857eqrdv 2619 . . . . . . . . 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 3621 . . . . . . 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 4500 . . . . . . . . 9 (𝑢 = ran 𝑓 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
6261sseq2d 3612 . . . . . . . 8 (𝑢 = ran 𝑓 → (𝑌 𝑥𝑢 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)))
6362rspcev 3295 . . . . . . 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 692 . . . . . 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 1860 . . . . 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 3025 . . . 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 202 . . 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 2980 . 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 268 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 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   ciun 4485  cmpt 4673   × cxp 5072  ran crn 5075  cres 5076   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  (class class class)co 6604  Fincfn 7899  +crp 11776  Metcme 19651  ballcbl 19652  TotBndctotbnd 33197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-2 11023  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-totbnd 33199
This theorem is referenced by:  totbndss  33208
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