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Theorem ssbnd 34130
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 10359 . . . . . . 7 0 ∈ ℝ
21ne0ii 4154 . . . . . 6 ℝ ≠ ∅
3 0ss 4198 . . . . . . . 8 ∅ ⊆ (𝑃(ball‘𝑀)𝑑)
4 sseq1 3852 . . . . . . . 8 (𝑌 = ∅ → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)))
53, 4mpbiri 250 . . . . . . 7 (𝑌 = ∅ → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
65ralrimivw 3177 . . . . . 6 (𝑌 = ∅ → ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7 r19.2z 4283 . . . . . 6 ((ℝ ≠ ∅ ∧ ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
82, 6, 7sylancr 583 . . . . 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 34125 . . . . . 6 ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) ↔ (𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
11 simplll 793 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑀 ∈ (Met‘𝑋))
12 ssbnd.2 . . . . . . . . . . . . . . . . . . . . 21 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
1312dmeqi 5558 . . . . . . . . . . . . . . . . . . . 20 dom 𝑁 = dom (𝑀 ↾ (𝑌 × 𝑌))
14 dmres 5656 . . . . . . . . . . . . . . . . . . . 20 dom (𝑀 ↾ (𝑌 × 𝑌)) = ((𝑌 × 𝑌) ∩ dom 𝑀)
1513, 14eqtri 2850 . . . . . . . . . . . . . . . . . . 19 dom 𝑁 = ((𝑌 × 𝑌) ∩ dom 𝑀)
16 xmetf 22505 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
1716fdmd 6288 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1815, 17syl5eqr 2876 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (∞Met‘𝑌) → ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
19 df-ss 3813 . . . . . . . . . . . . . . . . . 18 ((𝑌 × 𝑌) ⊆ dom 𝑀 ↔ ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
2018, 19sylibr 226 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (∞Met‘𝑌) → (𝑌 × 𝑌) ⊆ dom 𝑀)
2120ad2antlr 720 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 × 𝑌) ⊆ dom 𝑀)
22 metf 22506 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
2322fdmd 6288 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
2423ad3antrrr 723 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → dom 𝑀 = (𝑋 × 𝑋))
2521, 24sseqtrd 3867 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
26 dmss 5556 . . . . . . . . . . . . . . 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 5578 . . . . . . . . . . . . . 14 dom (𝑌 × 𝑌) = 𝑌
29 dmxpid 5578 . . . . . . . . . . . . . 14 dom (𝑋 × 𝑋) = 𝑋
3027, 28, 293sstr3g 3871 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑌𝑋)
31 simprl 789 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦𝑌)
3230, 31sseldd 3829 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦𝑋)
33 simpllr 795 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑃𝑋)
34 metcl 22508 . . . . . . . . . . . 12 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋𝑃𝑋) → (𝑦𝑀𝑃) ∈ ℝ)
3511, 32, 33, 34syl3anc 1496 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℝ)
36 rpre 12121 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
3736ad2antll 722 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
3835, 37readdcld 10387 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ)
39 metxmet 22510 . . . . . . . . . . . . 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 4026 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦 ∈ (𝑋𝑌))
42 rpxr 12124 . . . . . . . . . . . . 13 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
4342ad2antll 722 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
4412blres 22607 . . . . . . . . . . . 12 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
4540, 41, 43, 44syl3anc 1496 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
46 inss1 4058 . . . . . . . . . . . 12 ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑦(ball‘𝑀)𝑟)
4735leidd 10919 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (𝑦𝑀𝑃))
4835recnd 10386 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℂ)
4937recnd 10386 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℂ)
5048, 49pncand 10715 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (((𝑦𝑀𝑃) + 𝑟) − 𝑟) = (𝑦𝑀𝑃))
5147, 50breqtrrd 4902 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))
52 blss2 22580 . . . . . . . . . . . . 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 1510 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5446, 53syl5ss 3839 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5545, 54eqsstrd 3865 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
56 oveq2 6914 . . . . . . . . . . . 12 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → (𝑃(ball‘𝑀)𝑑) = (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5756sseq2d 3859 . . . . . . . . . . 11 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → ((𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))))
5857rspcev 3527 . . . . . . . . . 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 581 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
60 sseq1 3852 . . . . . . . . . 10 (𝑌 = (𝑦(ball‘𝑁)𝑟) → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
6160rexbidv 3263 . . . . . . . . 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 239 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6362rexlimdvva 3249 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6463expimpd 447 . . . . . 6 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6510, 64syl5bi 234 . . . . 5 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6665expdimp 446 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 ≠ ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
679, 66pm2.61dne 3086 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
6867ex 403 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (𝑁 ∈ (Bnd‘𝑌) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
69 simprr 791 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
70 xpss12 5358 . . . . . . 7 ((𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
7169, 69, 70syl2anc 581 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
7271resabs1d 5665 . . . . 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, 12syl6eqr 2880 . . . 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 34129 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
7539, 74syl3an1 1208 . . . . . . 7 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
76753expa 1153 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
7776adantrr 710 . . . . 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 34128 . . . . 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 581 . . . 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 2908 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑁 ∈ (Bnd‘𝑌))
8180rexlimdvaa 3242 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) → 𝑁 ∈ (Bnd‘𝑌)))
8268, 81impbid 204 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 198  wa 386   = wceq 1658  wcel 2166  wne 3000  wral 3118  wrex 3119  cin 3798  wss 3799  c0 4145   class class class wbr 4874   × cxp 5341  dom cdm 5343  cres 5345  cfv 6124  (class class class)co 6906  cr 10252  0cc0 10253   + caddc 10256  *cxr 10391  cle 10393  cmin 10586  +crp 12113  ∞Metcxmet 20092  Metcmet 20093  ballcbl 20094  Bndcbnd 34109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-po 5264  df-so 5265  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-er 8010  df-ec 8012  df-map 8125  df-en 8224  df-dom 8225  df-sdom 8226  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-div 11011  df-2 11415  df-rp 12114  df-xneg 12233  df-xadd 12234  df-xmul 12235  df-psmet 20099  df-xmet 20100  df-met 20101  df-bl 20102  df-bnd 34121
This theorem is referenced by:  prdsbnd2  34137  cntotbnd  34138
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