blelrn - Metamath Proof Explorer

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Theorem blelrn 23478

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 23468	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ^*)⟶𝒫 𝑋)
2	1	ffnd 6585	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) Fn (𝑋 × ℝ^*))
3		fnovrn 7425	. 2 ⊢ (((ball‘𝐷) Fn (𝑋 × ℝ^) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
4	2, 3	syl3an1 1161	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 1085 ∈ wcel 2108 𝒫 cpw 4530 × cxp 5578 ran crn 5581 Fn wfn 6413 ‘cfv 6418 (class class class)co 7255 ℝ^*cxr 10939 ∞Metcxmet 20495 ballcbl 20497

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-map 8575 df-xr 10944 df-psmet 20502 df-xmet 20503 df-bl 20505

This theorem is referenced by: unirnbl 23481 blssex 23488 blopn 23562 blcld 23567 metss 23570 metcnp3 23602 dscopn 23635 ioo2blex 23863

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