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Theorem ssbnd 35098
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 10617 . . . . . . 7 0 ∈ ℝ
21ne0ii 4275 . . . . . 6 ℝ ≠ ∅
3 0ss 4322 . . . . . . . 8 ∅ ⊆ (𝑃(ball‘𝑀)𝑑)
4 sseq1 3967 . . . . . . . 8 (𝑌 = ∅ → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)))
53, 4mpbiri 260 . . . . . . 7 (𝑌 = ∅ → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
65ralrimivw 3170 . . . . . 6 (𝑌 = ∅ → ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7 r19.2z 4412 . . . . . 6 ((ℝ ≠ ∅ ∧ ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
82, 6, 7sylancr 589 . . . . 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 35093 . . . . . 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 5745 . . . . . . . . . . . . . . . . . . . 20 dom 𝑁 = dom (𝑀 ↾ (𝑌 × 𝑌))
14 dmres 5847 . . . . . . . . . . . . . . . . . . . 20 dom (𝑀 ↾ (𝑌 × 𝑌)) = ((𝑌 × 𝑌) ∩ dom 𝑀)
1513, 14eqtri 2843 . . . . . . . . . . . . . . . . . . 19 dom 𝑁 = ((𝑌 × 𝑌) ∩ dom 𝑀)
16 xmetf 22911 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
1716fdmd 6495 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1815, 17syl5eqr 2869 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (∞Met‘𝑌) → ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
19 df-ss 3926 . . . . . . . . . . . . . . . . . 18 ((𝑌 × 𝑌) ⊆ dom 𝑀 ↔ ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
2018, 19sylibr 236 . . . . . . . . . . . . . . . . 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 22912 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
2322fdmd 6495 . . . . . . . . . . . . . . . . 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 3982 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
26 dmss 5743 . . . . . . . . . . . . . . 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 5772 . . . . . . . . . . . . . 14 dom (𝑌 × 𝑌) = 𝑌
29 dmxpid 5772 . . . . . . . . . . . . . 14 dom (𝑋 × 𝑋) = 𝑋
3027, 28, 293sstr3g 3986 . . . . . . . . . . . . 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 3943 . . . . . . . . . . . 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 22914 . . . . . . . . . . . 12 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋𝑃𝑋) → (𝑦𝑀𝑃) ∈ ℝ)
3511, 32, 33, 34syl3anc 1367 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℝ)
36 rpre 12372 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
3736ad2antll 727 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
3835, 37readdcld 10644 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ)
39 metxmet 22916 . . . . . . . . . . . . 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 4145 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦 ∈ (𝑋𝑌))
42 rpxr 12373 . . . . . . . . . . . . 13 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
4342ad2antll 727 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
4412blres 23013 . . . . . . . . . . . 12 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
4540, 41, 43, 44syl3anc 1367 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
46 inss1 4179 . . . . . . . . . . . 12 ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑦(ball‘𝑀)𝑟)
4735leidd 11180 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (𝑦𝑀𝑃))
4835recnd 10643 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℂ)
4937recnd 10643 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℂ)
5048, 49pncand 10972 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (((𝑦𝑀𝑃) + 𝑟) − 𝑟) = (𝑦𝑀𝑃))
5147, 50breqtrrd 5066 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))
52 blss2 22986 . . . . . . . . . . . . 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 1381 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5446, 53sstrid 3953 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5545, 54eqsstrd 3980 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
56 oveq2 7137 . . . . . . . . . . . 12 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → (𝑃(ball‘𝑀)𝑑) = (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5756sseq2d 3974 . . . . . . . . . . 11 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → ((𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))))
5857rspcev 3599 . . . . . . . . . 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 586 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
60 sseq1 3967 . . . . . . . . . 10 (𝑌 = (𝑦(ball‘𝑁)𝑟) → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
6160rexbidv 3282 . . . . . . . . 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 249 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6362rexlimdvva 3279 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6463expimpd 456 . . . . . 6 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6510, 64syl5bi 244 . . . . 5 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6665expdimp 455 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 ≠ ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
679, 66pm2.61dne 3092 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
6867ex 415 . 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 5542 . . . . . . 7 ((𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
7169, 69, 70syl2anc 586 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
7271resabs1d 5856 . . . . 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 2873 . . . 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 35097 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
7539, 74syl3an1 1159 . . . . . . 7 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
76753expa 1114 . . . . . 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 35096 . . . . 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 586 . . . 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 2912 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑁 ∈ (Bnd‘𝑌))
8180rexlimdvaa 3270 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) → 𝑁 ∈ (Bnd‘𝑌)))
8268, 81impbid 214 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 208  wa 398   = wceq 1537  wcel 2114  wne 3006  wral 3125  wrex 3126  cin 3908  wss 3909  c0 4265   class class class wbr 5038   × cxp 5525  dom cdm 5527  cres 5529  cfv 6327  (class class class)co 7129  cr 10510  0cc0 10511   + caddc 10514  *cxr 10648  cle 10650  cmin 10844  +crp 12364  ∞Metcxmet 20502  Metcmet 20503  ballcbl 20504  Bndcbnd 35077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435  ax-cnex 10567  ax-resscn 10568  ax-1cn 10569  ax-icn 10570  ax-addcl 10571  ax-addrcl 10572  ax-mulcl 10573  ax-mulrcl 10574  ax-mulcom 10575  ax-addass 10576  ax-mulass 10577  ax-distr 10578  ax-i2m1 10579  ax-1ne0 10580  ax-1rid 10581  ax-rnegex 10582  ax-rrecex 10583  ax-cnre 10584  ax-pre-lttri 10585  ax-pre-lttrn 10586  ax-pre-ltadd 10587  ax-pre-mulgt0 10588
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-po 5446  df-so 5447  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7087  df-ov 7132  df-oprab 7133  df-mpo 7134  df-1st 7663  df-2nd 7664  df-er 8263  df-ec 8265  df-map 8382  df-en 8484  df-dom 8485  df-sdom 8486  df-pnf 10651  df-mnf 10652  df-xr 10653  df-ltxr 10654  df-le 10655  df-sub 10846  df-neg 10847  df-div 11272  df-2 11675  df-rp 12365  df-xneg 12482  df-xadd 12483  df-xmul 12484  df-psmet 20509  df-xmet 20510  df-met 20511  df-bl 20512  df-bnd 35089
This theorem is referenced by:  prdsbnd2  35105  cntotbnd  35106
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