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Theorem bndss 36654
Description: A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
bndss ((𝑀 ∈ (Bndβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ (𝑀 β†Ύ (𝑆 Γ— 𝑆)) ∈ (Bndβ€˜π‘†))

Proof of Theorem bndss
Dummy variables π‘Ÿ π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metres2 23869 . . . 4 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ (𝑀 β†Ύ (𝑆 Γ— 𝑆)) ∈ (Metβ€˜π‘†))
21adantlr 714 . . 3 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) ∧ 𝑆 βŠ† 𝑋) β†’ (𝑀 β†Ύ (𝑆 Γ— 𝑆)) ∈ (Metβ€˜π‘†))
3 ssel2 3978 . . . . . . . . . . . . 13 ((𝑆 βŠ† 𝑋 ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑋)
43ancoms 460 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝑆 ∧ 𝑆 βŠ† 𝑋) β†’ π‘₯ ∈ 𝑋)
5 oveq1 7416 . . . . . . . . . . . . . . 15 (𝑦 = π‘₯ β†’ (𝑦(ballβ€˜π‘€)π‘Ÿ) = (π‘₯(ballβ€˜π‘€)π‘Ÿ))
65eqeq2d 2744 . . . . . . . . . . . . . 14 (𝑦 = π‘₯ β†’ (𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ) ↔ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ)))
76rexbidv 3179 . . . . . . . . . . . . 13 (𝑦 = π‘₯ β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ) ↔ βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ)))
87rspcva 3611 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝑋 ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ))
94, 8sylan 581 . . . . . . . . . . 11 (((π‘₯ ∈ 𝑆 ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ))
109adantlll 717 . . . . . . . . . 10 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ))
11 dfss 3967 . . . . . . . . . . . . . . . . . . 19 (𝑆 βŠ† 𝑋 ↔ 𝑆 = (𝑆 ∩ 𝑋))
1211biimpi 215 . . . . . . . . . . . . . . . . . 18 (𝑆 βŠ† 𝑋 β†’ 𝑆 = (𝑆 ∩ 𝑋))
13 incom 4202 . . . . . . . . . . . . . . . . . 18 (𝑆 ∩ 𝑋) = (𝑋 ∩ 𝑆)
1412, 13eqtrdi 2789 . . . . . . . . . . . . . . . . 17 (𝑆 βŠ† 𝑋 β†’ 𝑆 = (𝑋 ∩ 𝑆))
15 ineq1 4206 . . . . . . . . . . . . . . . . 17 (𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ) β†’ (𝑋 ∩ 𝑆) = ((π‘₯(ballβ€˜π‘€)π‘Ÿ) ∩ 𝑆))
1614, 15sylan9eq 2793 . . . . . . . . . . . . . . . 16 ((𝑆 βŠ† 𝑋 ∧ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ)) β†’ 𝑆 = ((π‘₯(ballβ€˜π‘€)π‘Ÿ) ∩ 𝑆))
1716adantll 713 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) ∧ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ)) β†’ 𝑆 = ((π‘₯(ballβ€˜π‘€)π‘Ÿ) ∩ 𝑆))
1817adantlr 714 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ)) β†’ 𝑆 = ((π‘₯(ballβ€˜π‘€)π‘Ÿ) ∩ 𝑆))
19 eqid 2733 . . . . . . . . . . . . . . . . . 18 (𝑀 β†Ύ (𝑆 Γ— 𝑆)) = (𝑀 β†Ύ (𝑆 Γ— 𝑆))
2019blssp 36624 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ∧ (π‘₯ ∈ 𝑆 ∧ π‘Ÿ ∈ ℝ+)) β†’ (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ) = ((π‘₯(ballβ€˜π‘€)π‘Ÿ) ∩ 𝑆))
2120an4s 659 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ (𝑆 βŠ† 𝑋 ∧ π‘Ÿ ∈ ℝ+)) β†’ (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ) = ((π‘₯(ballβ€˜π‘€)π‘Ÿ) ∩ 𝑆))
2221anassrs 469 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) ∧ π‘Ÿ ∈ ℝ+) β†’ (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ) = ((π‘₯(ballβ€˜π‘€)π‘Ÿ) ∩ 𝑆))
2322adantr 482 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ)) β†’ (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ) = ((π‘₯(ballβ€˜π‘€)π‘Ÿ) ∩ 𝑆))
2418, 23eqtr4d 2776 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ)) β†’ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ))
2524ex 414 . . . . . . . . . . . 12 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) ∧ π‘Ÿ ∈ ℝ+) β†’ (𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ) β†’ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ)))
2625reximdva 3169 . . . . . . . . . . 11 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ) β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ)))
2726imp 408 . . . . . . . . . 10 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) ∧ βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (π‘₯(ballβ€˜π‘€)π‘Ÿ)) β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ))
2810, 27syldan 592 . . . . . . . . 9 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ))
2928an32s 651 . . . . . . . 8 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) ∧ 𝑆 βŠ† 𝑋) β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ))
3029ex 414 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑆) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) β†’ (𝑆 βŠ† 𝑋 β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ)))
3130an32s 651 . . . . . 6 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) ∧ π‘₯ ∈ 𝑆) β†’ (𝑆 βŠ† 𝑋 β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ)))
3231imp 408 . . . . 5 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) ∧ π‘₯ ∈ 𝑆) ∧ 𝑆 βŠ† 𝑋) β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ))
3332an32s 651 . . . 4 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑆) β†’ βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ))
3433ralrimiva 3147 . . 3 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) ∧ 𝑆 βŠ† 𝑋) β†’ βˆ€π‘₯ ∈ 𝑆 βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ))
352, 34jca 513 . 2 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) ∧ 𝑆 βŠ† 𝑋) β†’ ((𝑀 β†Ύ (𝑆 Γ— 𝑆)) ∈ (Metβ€˜π‘†) ∧ βˆ€π‘₯ ∈ 𝑆 βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ)))
36 isbnd 36648 . . 3 (𝑀 ∈ (Bndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)))
3736anbi1i 625 . 2 ((𝑀 ∈ (Bndβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) ↔ ((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘Ÿ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)π‘Ÿ)) ∧ 𝑆 βŠ† 𝑋))
38 isbnd 36648 . 2 ((𝑀 β†Ύ (𝑆 Γ— 𝑆)) ∈ (Bndβ€˜π‘†) ↔ ((𝑀 β†Ύ (𝑆 Γ— 𝑆)) ∈ (Metβ€˜π‘†) ∧ βˆ€π‘₯ ∈ 𝑆 βˆƒπ‘Ÿ ∈ ℝ+ 𝑆 = (π‘₯(ballβ€˜(𝑀 β†Ύ (𝑆 Γ— 𝑆)))π‘Ÿ)))
3935, 37, 383imtr4i 292 1 ((𝑀 ∈ (Bndβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ (𝑀 β†Ύ (𝑆 Γ— 𝑆)) ∈ (Bndβ€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949   Γ— cxp 5675   β†Ύ cres 5679  β€˜cfv 6544  (class class class)co 7409  β„+crp 12974  Metcmet 20930  ballcbl 20931  Bndcbnd 36635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-mulcl 11172  ax-i2m1 11178
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-rp 12975  df-xadd 13093  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-bnd 36647
This theorem is referenced by:  ssbnd  36656
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