Theorem blrnps 24383

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.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
blrnps	⊢ (𝐷 ∈ (PsMet‘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 blrnps

Step	Hyp	Ref	Expression
1		blfps 24381	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
2		ffn 6662	. 2 ⊢ ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 → (ball‘𝐷) Fn (𝑋 × ℝ*))
3		ovelrn 7536	. 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 ∈ (PsMet‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  𝒫 cpw 4542   × cxp 5622  ran crn 5625   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  ℝ^*cxr 11169  PsMetcpsmet 21328  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-bl 21339

This theorem is referenced by:  blssps  24399