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Theorem ssbnd 37775
Description: A subset of a metric space is bounded iff it is contained in a ball around 𝑃, for any 𝑃 in the larger space. (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypothesis
Ref Expression
ssbnd.2 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
Assertion
Ref Expression
ssbnd ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (𝑁 ∈ (Bnd‘𝑌) ↔ ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
Distinct variable groups:   𝑀,𝑑   𝑁,𝑑   𝑃,𝑑   𝑋,𝑑   𝑌,𝑑

Proof of Theorem ssbnd
Dummy variables 𝑟 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0re 11261 . . . . . . 7 0 ∈ ℝ
21ne0ii 4350 . . . . . 6 ℝ ≠ ∅
3 0ss 4406 . . . . . . . 8 ∅ ⊆ (𝑃(ball‘𝑀)𝑑)
4 sseq1 4021 . . . . . . . 8 (𝑌 = ∅ → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)))
53, 4mpbiri 258 . . . . . . 7 (𝑌 = ∅ → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
65ralrimivw 3148 . . . . . 6 (𝑌 = ∅ → ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7 r19.2z 4501 . . . . . 6 ((ℝ ≠ ∅ ∧ ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
82, 6, 7sylancr 587 . . . . 5 (𝑌 = ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
98a1i 11 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 = ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
10 isbnd2 37770 . . . . . 6 ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) ↔ (𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
11 simplll 775 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑀 ∈ (Met‘𝑋))
12 ssbnd.2 . . . . . . . . . . . . . . . . . . . . 21 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
1312dmeqi 5918 . . . . . . . . . . . . . . . . . . . 20 dom 𝑁 = dom (𝑀 ↾ (𝑌 × 𝑌))
14 dmres 6032 . . . . . . . . . . . . . . . . . . . 20 dom (𝑀 ↾ (𝑌 × 𝑌)) = ((𝑌 × 𝑌) ∩ dom 𝑀)
1513, 14eqtri 2763 . . . . . . . . . . . . . . . . . . 19 dom 𝑁 = ((𝑌 × 𝑌) ∩ dom 𝑀)
16 xmetf 24355 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
1716fdmd 6747 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1815, 17eqtr3id 2789 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (∞Met‘𝑌) → ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
19 dfss2 3981 . . . . . . . . . . . . . . . . . 18 ((𝑌 × 𝑌) ⊆ dom 𝑀 ↔ ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
2018, 19sylibr 234 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (∞Met‘𝑌) → (𝑌 × 𝑌) ⊆ dom 𝑀)
2120ad2antlr 727 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 × 𝑌) ⊆ dom 𝑀)
22 metf 24356 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
2322fdmd 6747 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
2423ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → dom 𝑀 = (𝑋 × 𝑋))
2521, 24sseqtrd 4036 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
26 dmss 5916 . . . . . . . . . . . . . . 15 ((𝑌 × 𝑌) ⊆ (𝑋 × 𝑋) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
2725, 26syl 17 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
28 dmxpid 5944 . . . . . . . . . . . . . 14 dom (𝑌 × 𝑌) = 𝑌
29 dmxpid 5944 . . . . . . . . . . . . . 14 dom (𝑋 × 𝑋) = 𝑋
3027, 28, 293sstr3g 4040 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑌𝑋)
31 simprl 771 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦𝑌)
3230, 31sseldd 3996 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦𝑋)
33 simpllr 776 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑃𝑋)
34 metcl 24358 . . . . . . . . . . . 12 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋𝑃𝑋) → (𝑦𝑀𝑃) ∈ ℝ)
3511, 32, 33, 34syl3anc 1370 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℝ)
36 rpre 13041 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
3736ad2antll 729 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
3835, 37readdcld 11288 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ)
39 metxmet 24360 . . . . . . . . . . . . 13 (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
4011, 39syl 17 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑀 ∈ (∞Met‘𝑋))
4132, 31elind 4210 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦 ∈ (𝑋𝑌))
42 rpxr 13042 . . . . . . . . . . . . 13 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
4342ad2antll 729 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
4412blres 24457 . . . . . . . . . . . 12 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
4540, 41, 43, 44syl3anc 1370 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
46 inss1 4245 . . . . . . . . . . . 12 ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑦(ball‘𝑀)𝑟)
4735leidd 11827 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (𝑦𝑀𝑃))
4835recnd 11287 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℂ)
4937recnd 11287 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℂ)
5048, 49pncand 11619 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (((𝑦𝑀𝑃) + 𝑟) − 𝑟) = (𝑦𝑀𝑃))
5147, 50breqtrrd 5176 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))
52 blss2 24430 . . . . . . . . . . . . 13 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋𝑃𝑋) ∧ (𝑟 ∈ ℝ ∧ ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ ∧ (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5340, 32, 33, 37, 38, 51, 52syl33anc 1384 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5446, 53sstrid 4007 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5545, 54eqsstrd 4034 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
56 oveq2 7439 . . . . . . . . . . . 12 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → (𝑃(ball‘𝑀)𝑑) = (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5756sseq2d 4028 . . . . . . . . . . 11 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → ((𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))))
5857rspcev 3622 . . . . . . . . . 10 ((((𝑦𝑀𝑃) + 𝑟) ∈ ℝ ∧ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
5938, 55, 58syl2anc 584 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
60 sseq1 4021 . . . . . . . . . 10 (𝑌 = (𝑦(ball‘𝑁)𝑟) → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
6160rexbidv 3177 . . . . . . . . 9 (𝑌 = (𝑦(ball‘𝑁)𝑟) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
6259, 61syl5ibrcom 247 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6362rexlimdvva 3211 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6463expimpd 453 . . . . . 6 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6510, 64biimtrid 242 . . . . 5 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6665expdimp 452 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 ≠ ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
679, 66pm2.61dne 3026 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
6867ex 412 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (𝑁 ∈ (Bnd‘𝑌) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
69 simprr 773 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
70 xpss12 5704 . . . . . . 7 ((𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
7169, 69, 70syl2anc 584 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
7271resabs1d 6028 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → ((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ↾ (𝑌 × 𝑌)) = (𝑀 ↾ (𝑌 × 𝑌)))
7372, 12eqtr4di 2793 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → ((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ↾ (𝑌 × 𝑌)) = 𝑁)
74 blbnd 37774 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
7539, 74syl3an1 1162 . . . . . . 7 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
76753expa 1117 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
7776adantrr 717 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
78 bndss 37773 . . . . 5 (((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)) ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ↾ (𝑌 × 𝑌)) ∈ (Bnd‘𝑌))
7977, 69, 78syl2anc 584 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → ((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ↾ (𝑌 × 𝑌)) ∈ (Bnd‘𝑌))
8073, 79eqeltrrd 2840 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑁 ∈ (Bnd‘𝑌))
8180rexlimdvaa 3154 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) → 𝑁 ∈ (Bnd‘𝑌)))
8268, 81impbid 212 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (𝑁 ∈ (Bnd‘𝑌) ↔ ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cin 3962  wss 3963  c0 4339   class class class wbr 5148   × cxp 5687  dom cdm 5689  cres 5691  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153   + caddc 11156  *cxr 11292  cle 11294  cmin 11490  +crp 13032  ∞Metcxmet 21367  Metcmet 21368  ballcbl 21369  Bndcbnd 37754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-ec 8746  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-2 12327  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-bnd 37766
This theorem is referenced by:  prdsbnd2  37782  cntotbnd  37783
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