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Theorem blrn 23021

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 23019	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ^*)⟶𝒫 𝑋)
2		ffn 6516	. 2 ⊢ ((ball‘𝐷):(𝑋 × ℝ^)⟶𝒫 𝑋 → (ball‘𝐷) Fn (𝑋 × ℝ^))
3		ovelrn 7326	. 2 ⊢ ((ball‘𝐷) Fn (𝑋 × ℝ^) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ^ 𝐴 = (𝑥(ball‘𝐷)𝑟)))
4	1, 2, 3	3syl 18	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 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 𝒫 cpw 4541 × cxp 5555 ran crn 5558 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝ^*cxr 10676 ∞Metcxmet 20532 ballcbl 20534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-xr 10681 df-psmet 20539 df-xmet 20540 df-bl 20542

This theorem is referenced by: blss 23037 imasf1oxms 23101 prdsxmslem2 23141 blssioo 23405

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