Theorem unirnblps 13216

Description: The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
unirnblps	⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)

Proof of Theorem unirnblps

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		blfps 13203	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
2	1	frnd 5357	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
3		sspwuni 3957	. . 3 ⊢ (ran (ball‘𝐷) ⊆ 𝒫 𝑋 ↔ ∪ ran (ball‘𝐷) ⊆ 𝑋)
4	2, 3	sylib 121	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) ⊆ 𝑋)
5		1rp 9614	. . . 4 ⊢ 1 ∈ ℝ+
6		blcntrps 13209	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐷)1))
7	5, 6	mp3an3 1321	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (𝑥(ball‘𝐷)1))
8		rpxr 9618	. . . . 5 ⊢ (1 ∈ ℝ+ → 1 ∈ ℝ*)
9	5, 8	ax-mp 5	. . . 4 ⊢ 1 ∈ ℝ*
10		blelrnps 13213	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ*) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷))
11	9, 10	mp3an3 1321	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷))
12		elunii 3801	. . 3 ⊢ ((𝑥 ∈ (𝑥(ball‘𝐷)1) ∧ (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) → 𝑥 ∈ ∪ ran (ball‘𝐷))
13	7, 11, 12	syl2anc 409	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ ran (ball‘𝐷))
14	4, 13	eqelssd 3166	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141   ⊆ wss 3121  𝒫 cpw 3566  ∪ cuni 3796   × cxp 4609  ran crn 4612  ‘cfv 5198  (class class class)co 5853  1c1 7775  ℝ^*cxr 7953  ℝ⁺crp 9610  PsMetcpsmet 12773  ballcbl 12776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871  ax-0lt1 7880  ax-rnegex 7883

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-rp 9611  df-psmet 12781  df-bl 12784

This theorem is referenced by: (None)