blelrn - Metamath Proof Explorer

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Theorem blelrn 24392

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 24382	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ^*)⟶𝒫 𝑋)
2	1	ffnd 6663	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) Fn (𝑋 × ℝ^*))
3		fnovrn 7535	. 2 ⊢ (((ball‘𝐷) Fn (𝑋 × ℝ^) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
4	2, 3	syl3an1 1164	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 1087 ∈ wcel 2114 𝒫 cpw 4542 × cxp 5622 ran crn 5625 Fn wfn 6487 ‘cfv 6492 (class class class)co 7360 ℝ^*cxr 11169 ∞Metcxmet 21329 ballcbl 21331

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-map 8768 df-xr 11174 df-psmet 21336 df-xmet 21337 df-bl 21339

This theorem is referenced by: unirnbl 24395 blssex 24402 blopn 24475 blcld 24480 metss 24483 metcnp3 24515 dscopn 24548 ioo2blex 24769

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