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Theorem blrn 24392

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 24390	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ^*)⟶𝒫 𝑋)
2		ffn 6655	. 2 ⊢ ((ball‘𝐷):(𝑋 × ℝ^)⟶𝒫 𝑋 → (ball‘𝐷) Fn (𝑋 × ℝ^))
3		ovelrn 7532	. 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 207 = wceq 1547 ∈ wcel 2119 ∃wrex 3063 𝒫 cpw 4529 × cxp 5616 ran crn 5619 Fn wfn 6480 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ℝ^*cxr 11169 ∞Metcxmet 21332 ballcbl 21334

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-map 8765 df-xr 11174 df-psmet 21339 df-xmet 21340 df-bl 21342

This theorem is referenced by: blss 24408 imasf1oxms 24472 prdsxmslem2 24512 blssioo 24778

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