blelrn - Metamath Proof Explorer

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Theorem blelrn 22591

Description: A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

**Assertion**
Ref	Expression
blelrn	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))

**Proof of Theorem blelrn**
Step	Hyp	Ref	Expression
1		blf 22581	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ^*)⟶𝒫 𝑋)
2	1	ffnd 6278	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) Fn (𝑋 × ℝ^*))
3		fnovrn 7068	. 2 ⊢ (((ball‘𝐷) Fn (𝑋 × ℝ^) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
4	2, 3	syl3an1 1208	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1113 ∈ wcel 2166 𝒫 cpw 4377 × cxp 5339 ran crn 5342 Fn wfn 6117 ‘cfv 6122 (class class class)co 6904 ℝ^*cxr 10389 ∞Metcxmet 20090 ballcbl 20092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-fv 6130 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-1st 7427 df-2nd 7428 df-map 8123 df-xr 10394 df-psmet 20097 df-xmet 20098 df-bl 20100

This theorem is referenced by: unirnbl 22594 blssex 22601 blopn 22674 blcld 22679 metss 22682 metcnp3 22714 dscopn 22747 ioo2blex 22966

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