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Theorem ssbnd 33254
 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 9992 . . . . . . 7 0 ∈ ℝ
21ne0ii 3904 . . . . . 6 ℝ ≠ ∅
3 0ss 3949 . . . . . . . 8 ∅ ⊆ (𝑃(ball‘𝑀)𝑑)
4 sseq1 3610 . . . . . . . 8 (𝑌 = ∅ → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)))
53, 4mpbiri 248 . . . . . . 7 (𝑌 = ∅ → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
65ralrimivw 2962 . . . . . 6 (𝑌 = ∅ → ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7 r19.2z 4037 . . . . . 6 ((ℝ ≠ ∅ ∧ ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
82, 6, 7sylancr 694 . . . . 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 33249 . . . . . 6 ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) ↔ (𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
11 simplll 797 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑀 ∈ (Met‘𝑋))
12 ssbnd.2 . . . . . . . . . . . . . . . . . . . . 21 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
1312dmeqi 5290 . . . . . . . . . . . . . . . . . . . 20 dom 𝑁 = dom (𝑀 ↾ (𝑌 × 𝑌))
14 dmres 5383 . . . . . . . . . . . . . . . . . . . 20 dom (𝑀 ↾ (𝑌 × 𝑌)) = ((𝑌 × 𝑌) ∩ dom 𝑀)
1513, 14eqtri 2643 . . . . . . . . . . . . . . . . . . 19 dom 𝑁 = ((𝑌 × 𝑌) ∩ dom 𝑀)
16 xmetf 22057 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
17 fdm 6013 . . . . . . . . . . . . . . . . . . . 20 (𝑁:(𝑌 × 𝑌)⟶ℝ* → dom 𝑁 = (𝑌 × 𝑌))
1816, 17syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1915, 18syl5eqr 2669 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (∞Met‘𝑌) → ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
20 df-ss 3573 . . . . . . . . . . . . . . . . . 18 ((𝑌 × 𝑌) ⊆ dom 𝑀 ↔ ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
2119, 20sylibr 224 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (∞Met‘𝑌) → (𝑌 × 𝑌) ⊆ dom 𝑀)
2221ad2antlr 762 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 × 𝑌) ⊆ dom 𝑀)
23 metf 22058 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
24 fdm 6013 . . . . . . . . . . . . . . . . . 18 (𝑀:(𝑋 × 𝑋)⟶ℝ → dom 𝑀 = (𝑋 × 𝑋))
2523, 24syl 17 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
2625ad3antrrr 765 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → dom 𝑀 = (𝑋 × 𝑋))
2722, 26sseqtrd 3625 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
28 dmss 5288 . . . . . . . . . . . . . . 15 ((𝑌 × 𝑌) ⊆ (𝑋 × 𝑋) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
2927, 28syl 17 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
30 dmxpid 5310 . . . . . . . . . . . . . 14 dom (𝑌 × 𝑌) = 𝑌
31 dmxpid 5310 . . . . . . . . . . . . . 14 dom (𝑋 × 𝑋) = 𝑋
3229, 30, 313sstr3g 3629 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑌𝑋)
33 simprl 793 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦𝑌)
3432, 33sseldd 3588 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦𝑋)
35 simpllr 798 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑃𝑋)
36 metcl 22060 . . . . . . . . . . . 12 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦𝑋𝑃𝑋) → (𝑦𝑀𝑃) ∈ ℝ)
3711, 34, 35, 36syl3anc 1323 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℝ)
38 rpre 11791 . . . . . . . . . . . 12 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
3938ad2antll 764 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
4037, 39readdcld 10021 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ)
41 metxmet 22062 . . . . . . . . . . . . 13 (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
4211, 41syl 17 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑀 ∈ (∞Met‘𝑋))
4334, 33elind 3781 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑦 ∈ (𝑋𝑌))
44 rpxr 11792 . . . . . . . . . . . . 13 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
4544ad2antll 764 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
4612blres 22159 . . . . . . . . . . . 12 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
4742, 43, 45, 46syl3anc 1323 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
48 inss1 3816 . . . . . . . . . . . 12 ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑦(ball‘𝑀)𝑟)
4937leidd 10546 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (𝑦𝑀𝑃))
5037recnd 10020 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ∈ ℂ)
5139recnd 10020 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → 𝑟 ∈ ℂ)
5250, 51pncand 10345 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (((𝑦𝑀𝑃) + 𝑟) − 𝑟) = (𝑦𝑀𝑃))
5349, 52breqtrrd 4646 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))
54 blss2 22132 . . . . . . . . . . . . 13 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋𝑃𝑋) ∧ (𝑟 ∈ ℝ ∧ ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ ∧ (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5542, 34, 35, 39, 40, 53, 54syl33anc 1338 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5648, 55syl5ss 3598 . . . . . . . . . . 11 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5747, 56eqsstrd 3623 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
58 oveq2 6618 . . . . . . . . . . . 12 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → (𝑃(ball‘𝑀)𝑑) = (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
5958sseq2d 3617 . . . . . . . . . . 11 (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → ((𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))))
6059rspcev 3298 . . . . . . . . . 10 ((((𝑦𝑀𝑃) + 𝑟) ∈ ℝ ∧ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
6140, 57, 60syl2anc 692 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
62 sseq1 3610 . . . . . . . . . 10 (𝑌 = (𝑦(ball‘𝑁)𝑟) → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
6362rexbidv 3046 . . . . . . . . 9 (𝑌 = (𝑦(ball‘𝑁)𝑟) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
6461, 63syl5ibrcom 237 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦𝑌𝑟 ∈ ℝ+)) → (𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6564rexlimdvva 3032 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6665expimpd 628 . . . . . 6 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦𝑌𝑟 ∈ ℝ+ 𝑌 = (𝑦(ball‘𝑁)𝑟)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6710, 66syl5bi 232 . . . . 5 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
6867expdimp 453 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 ≠ ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
699, 68pm2.61dne 2876 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7069ex 450 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (𝑁 ∈ (Bnd‘𝑌) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
71 simprr 795 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
72 xpss12 5191 . . . . . . 7 ((𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
7371, 71, 72syl2anc 692 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
7473resabs1d 5392 . . . . 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)))
7574, 12syl6eqr 2673 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → ((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ↾ (𝑌 × 𝑌)) = 𝑁)
76 blbnd 33253 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
7741, 76syl3an1 1356 . . . . . . 7 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
78773expa 1262 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
7978adantrr 752 . . . . 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)))
80 bndss 33252 . . . . 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))
8179, 71, 80syl2anc 692 . . . 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))
8275, 81eqeltrrd 2699 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑁 ∈ (Bnd‘𝑌))
8382rexlimdvaa 3026 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) → 𝑁 ∈ (Bnd‘𝑌)))
8470, 83impbid 202 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 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908   ∩ cin 3558   ⊆ wss 3559  ∅c0 3896   class class class wbr 4618   × cxp 5077  dom cdm 5079   ↾ cres 5081  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610  ℝcr 9887  0cc0 9888   + caddc 9891  ℝ*cxr 10025   ≤ cle 10027   − cmin 10218  ℝ+crp 11784  ∞Metcxmt 19663  Metcme 19664  ballcbl 19665  Bndcbnd 33233 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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965 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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-er 7694  df-ec 7696  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-2 11031  df-rp 11785  df-xneg 11898  df-xadd 11899  df-xmul 11900  df-psmet 19670  df-xmet 19671  df-met 19672  df-bl 19673  df-bnd 33245 This theorem is referenced by:  prdsbnd2  33261  cntotbnd  33262
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