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Theorem blrn 24440

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 24438	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ^*)⟶𝒫 𝑋)
2		ffn 6747	. 2 ⊢ ((ball‘𝐷):(𝑋 × ℝ^)⟶𝒫 𝑋 → (ball‘𝐷) Fn (𝑋 × ℝ^))
3		ovelrn 7626	. 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 206 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 𝒫 cpw 4622 × cxp 5698 ran crn 5701 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℝ^*cxr 11323 ∞Metcxmet 21372 ballcbl 21374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-map 8886 df-xr 11328 df-psmet 21379 df-xmet 21380 df-bl 21382

This theorem is referenced by: blss 24456 imasf1oxms 24523 prdsxmslem2 24563 blssioo 24836

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