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Theorem ssbnd 36247
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 11157 . . . . . . 7 0 ∈ ℝ
21ne0ii 4297 . . . . . 6 ℝ ≠ ∅
3 0ss 4356 . . . . . . . 8 ∅ ⊆ (𝑃(ball‘𝑀)𝑑)
4 sseq1 3969 . . . . . . . 8 (𝑌 = ∅ → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)))
53, 4mpbiri 257 . . . . . . 7 (𝑌 = ∅ → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
65ralrimivw 3147 . . . . . 6 (𝑌 = ∅ → ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7 r19.2z 4452 . . . . . 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 36242 . . . . . 6 ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) ↔ (𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
11 simplll 773 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑀 ∈ (Met‘𝑋))
12 ssbnd.2 . . . . . . . . . . . . . . . . . . . . 21 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
1312dmeqi 5860 . . . . . . . . . . . . . . . . . . . 20 dom 𝑁 = dom (𝑀 ↾ (𝑌 × 𝑌))
14 dmres 5959 . . . . . . . . . . . . . . . . . . . 20 dom (𝑀 ↾ (𝑌 × 𝑌)) = ((𝑌 × 𝑌) ∩ dom 𝑀)
1513, 14eqtri 2764 . . . . . . . . . . . . . . . . . . 19 dom 𝑁 = ((𝑌 × 𝑌) ∩ dom 𝑀)
16 xmetf 23682 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
1716fdmd 6679 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1815, 17eqtr3id 2790 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (∞Met‘𝑌) → ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
19 df-ss 3927 . . . . . . . . . . . . . . . . . 18 ((𝑌 × 𝑌) ⊆ dom 𝑀 ↔ ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
2018, 19sylibr 233 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (∞Met‘𝑌) → (𝑌 × 𝑌) ⊆ dom 𝑀)
2120ad2antlr 725 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 × 𝑌) ⊆ dom 𝑀)
22 metf 23683 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
2322fdmd 6679 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
2423ad3antrrr 728 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → dom 𝑀 = (𝑋 × 𝑋))
2521, 24sseqtrd 3984 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
26 dmss 5858 . . . . . . . . . . . . . . 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 5885 . . . . . . . . . . . . . 14 dom (𝑌 × 𝑌) = 𝑌
29 dmxpid 5885 . . . . . . . . . . . . . 14 dom (𝑋 × 𝑋) = 𝑋
3027, 28, 293sstr3g 3988 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑌𝑋)
31 simprl 769 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦𝑌)
3230, 31sseldd 3945 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦𝑋)
33 simpllr 774 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑃𝑋)
34 metcl 23685 . . . . . . . . . . . 12 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋𝑃𝑋) → (𝑦𝑀𝑃) ∈ ℝ)
3511, 32, 33, 34syl3anc 1371 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℝ)
36 rpre 12923 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
3736ad2antll 727 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
3835, 37readdcld 11184 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ)
39 metxmet 23687 . . . . . . . . . . . . 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 4154 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦 ∈ (𝑋𝑌))
42 rpxr 12924 . . . . . . . . . . . . 13 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
4342ad2antll 727 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
4412blres 23784 . . . . . . . . . . . 12 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
4540, 41, 43, 44syl3anc 1371 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
46 inss1 4188 . . . . . . . . . . . 12 ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑦(ball‘𝑀)𝑟)
4735leidd 11721 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (𝑦𝑀𝑃))
4835recnd 11183 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℂ)
4937recnd 11183 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℂ)
5048, 49pncand 11513 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (((𝑦𝑀𝑃) + 𝑟) − 𝑟) = (𝑦𝑀𝑃))
5147, 50breqtrrd 5133 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))
52 blss2 23757 . . . . . . . . . . . . 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 1385 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5446, 53sstrid 3955 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5545, 54eqsstrd 3982 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
56 oveq2 7365 . . . . . . . . . . . 12 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → (𝑃(ball‘𝑀)𝑑) = (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5756sseq2d 3976 . . . . . . . . . . 11 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → ((𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))))
5857rspcev 3581 . . . . . . . . . 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 3969 . . . . . . . . . 10 (𝑌 = (𝑦(ball‘𝑁)𝑟) → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
6160rexbidv 3175 . . . . . . . . 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 246 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6362rexlimdvva 3205 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6463expimpd 454 . . . . . 6 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6510, 64biimtrid 241 . . . . 5 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6665expdimp 453 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 ≠ ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
679, 66pm2.61dne 3031 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
6867ex 413 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (𝑁 ∈ (Bnd‘𝑌) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
69 simprr 771 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
70 xpss12 5648 . . . . . . 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 5968 . . . . 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 2794 . . . 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 36246 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
7539, 74syl3an1 1163 . . . . . . 7 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
76753expa 1118 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
7776adantrr 715 . . . . 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 36245 . . . . 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 2839 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑁 ∈ (Bnd‘𝑌))
8180rexlimdvaa 3153 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) → 𝑁 ∈ (Bnd‘𝑌)))
8268, 81impbid 211 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 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  cin 3909  wss 3910  c0 4282   class class class wbr 5105   × cxp 5631  dom cdm 5633  cres 5635  cfv 6496  (class class class)co 7357  cr 11050  0cc0 11051   + caddc 11054  *cxr 11188  cle 11190  cmin 11385  +crp 12915  ∞Metcxmet 20781  Metcmet 20782  ballcbl 20783  Bndcbnd 36226
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-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-id 5531  df-po 5545  df-so 5546  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-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-1st 7921  df-2nd 7922  df-er 8648  df-ec 8650  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  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-bnd 36238
This theorem is referenced by:  prdsbnd2  36254  cntotbnd  36255
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