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Mirrors > Home > MPE Home > Th. List > Mathboxes > bndmet | Structured version Visualization version GIF version |
Description: A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
bndmet | ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isbnd 36585 | . 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑦))) | |
2 | 1 | simplbi 499 | 1 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ‘cfv 6539 (class class class)co 7403 ℝ+crp 12969 Metcmet 20914 ballcbl 20915 Bndcbnd 36572 |
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 5297 ax-nul 5304 ax-pr 5425 |
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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6491 df-fun 6541 df-fv 6547 df-ov 7406 df-bnd 36584 |
This theorem is referenced by: isbnd3 36589 equivbnd 36595 bnd2lem 36596 equivbnd2 36597 prdsbnd 36598 prdsbnd2 36600 |
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