Theorem unirnblps 15009

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 14996	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
2	1	frnd 5455	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) ⊆ 𝒫 𝑋)
3		sspwuni 4026	. . 3 ⊢ (ran (ball‘𝐷) ⊆ 𝒫 𝑋 ↔ ∪ ran (ball‘𝐷) ⊆ 𝑋)
4	2, 3	sylib 122	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) ⊆ 𝑋)
5		1rp 9814	. . . 4 ⊢ 1 ∈ ℝ+
6		blcntrps 15002	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐷)1))
7	5, 6	mp3an3 1339	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (𝑥(ball‘𝐷)1))
8		rpxr 9818	. . . . 5 ⊢ (1 ∈ ℝ+ → 1 ∈ ℝ*)
9	5, 8	ax-mp 5	. . . 4 ⊢ 1 ∈ ℝ*
10		blelrnps 15006	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ*) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷))
11	9, 10	mp3an3 1339	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷))
12		elunii 3869	. . 3 ⊢ ((𝑥 ∈ (𝑥(ball‘𝐷)1) ∧ (𝑥(ball‘𝐷)1) ∈ ran (ball‘𝐷)) → 𝑥 ∈ ∪ ran (ball‘𝐷))
13	7, 11, 12	syl2anc 411	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ ran (ball‘𝐷))
14	4, 13	eqelssd 3220	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2178   ⊆ wss 3174  𝒫 cpw 3626  ∪ cuni 3864   × cxp 4691  ran crn 4694  ‘cfv 5290  (class class class)co 5967  1c1 7961  ℝ^*cxr 8141  ℝ⁺crp 9810  PsMetcpsmet 14412  ballcbl 14415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057  ax-0lt1 8066  ax-rnegex 8069

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-rp 9811  df-psmet 14420  df-bl 14423

This theorem is referenced by: (None)