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Theorem blrn 24531

Description: Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

**Assertion**
Ref	Expression
blrn	⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ^* 𝐴 = (𝑥(ball‘𝐷)𝑟)))

Distinct variable groups: 𝑥,𝑟,𝐴 𝐷,𝑟,𝑥 𝑋,𝑟,𝑥

**Proof of Theorem blrn**
Step	Hyp	Ref	Expression
1		blf 24529	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ^*)⟶𝒫 𝑋)
2		ffn 6703	. 2 ⊢ ((ball‘𝐷):(𝑋 × ℝ^)⟶𝒫 𝑋 → (ball‘𝐷) Fn (𝑋 × ℝ^))
3		ovelrn 7584	. 2 ⊢ ((ball‘𝐷) Fn (𝑋 × ℝ^) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ^ 𝐴 = (𝑥(ball‘𝐷)𝑟)))
4	1, 2, 3	3syl 19	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ^* 𝐴 = (𝑥(ball‘𝐷)𝑟)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 = wceq 1567 ∈ wcel 2149 ∃wrex 3095 𝒫 cpw 4564 × cxp 5657 ran crn 5660 Fn wfn 6529 ⟶wf 6530 ‘cfv 6534 (class class class)co 7408 ℝ^*cxr 11238 ∞Metcxmet 21472 ballcbl 21474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7982 df-2nd 7983 df-map 8822 df-xr 11243 df-psmet 21479 df-xmet 21480 df-bl 21482

This theorem is referenced by: blss 24547 imasf1oxms 24611 prdsxmslem2 24651 blssioo 24917

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