blelrn - Metamath Proof Explorer

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Theorem blelrn 23570

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 23560	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ^*)⟶𝒫 𝑋)
2	1	ffnd 6601	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) Fn (𝑋 × ℝ^*))
3		fnovrn 7447	. 2 ⊢ (((ball‘𝐷) Fn (𝑋 × ℝ^) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
4	2, 3	syl3an1 1162	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 1086 ∈ wcel 2106 𝒫 cpw 4533 × cxp 5587 ran crn 5590 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 ℝ^*cxr 11008 ∞Metcxmet 20582 ballcbl 20584

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-map 8617 df-xr 11013 df-psmet 20589 df-xmet 20590 df-bl 20592

This theorem is referenced by: unirnbl 23573 blssex 23580 blopn 23656 blcld 23661 metss 23664 metcnp3 23696 dscopn 23729 ioo2blex 23957

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