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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 35966 | . 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑦))) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2101 ∀wral 3059 ∃wrex 3068 ‘cfv 6447 (class class class)co 7295 ℝ+crp 12758 Metcmet 20611 ballcbl 20612 Bndcbnd 35953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3224 df-v 3436 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-iota 6399 df-fun 6449 df-fv 6455 df-ov 7298 df-bnd 35965 |
This theorem is referenced by: isbnd3 35970 equivbnd 35976 bnd2lem 35977 equivbnd2 35978 prdsbnd 35979 prdsbnd2 35981 |
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