Theorem metustfbas 24591

Description: The filter base generated by a metric 𝐷. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Hypothesis

Ref	Expression
metust.1	⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))

Assertion

Ref	Expression
metustfbas	⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))

Distinct variable groups:   𝐷,𝑎   𝑋,𝑎   𝐹,𝑎

Proof of Theorem metustfbas

Dummy variables 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metust.1	. . . . . . 7 ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
2	1	metustel 24584	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎))))
3		simpr 484	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → 𝑥 = (◡𝐷 “ (0[,)𝑎)))
4		cnvimass 6111	. . . . . . . . . 10 ⊢ (◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
5		psmetf 24337	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	fdmd 6757	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
7	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → dom 𝐷 = (𝑋 × 𝑋))
8	4, 7	sseqtrid 4061	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
9	3, 8	eqsstrd 4047	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → 𝑥 ⊆ (𝑋 × 𝑋))
10	9	ex 412	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 = (◡𝐷 “ (0[,)𝑎)) → 𝑥 ⊆ (𝑋 × 𝑋)))
11	10	rexlimdvw 3166	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)) → 𝑥 ⊆ (𝑋 × 𝑋)))
12	2, 11	sylbid 240	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ (𝑋 × 𝑋)))
13	12	ralrimiv 3151	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝐹 𝑥 ⊆ (𝑋 × 𝑋))
14		pwssb 5124	. . . 4 ⊢ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑥 ∈ 𝐹 𝑥 ⊆ (𝑋 × 𝑋))
15	13, 14	sylibr 234	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
16	15	adantl 481	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
17		cnvexg 7964	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
18		imaexg 7953	. . . . . . 7 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)1)) ∈ V)
19		elisset 2826	. . . . . . 7 ⊢ ((◡𝐷 “ (0[,)1)) ∈ V → ∃𝑥 𝑥 = (◡𝐷 “ (0[,)1)))
20		1rp 13061	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
21		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → (0[,)𝑎) = (0[,)1))
22	21	imaeq2d 6089	. . . . . . . . . 10 ⊢ (𝑎 = 1 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)1)))
23	22	rspceeqv 3658	. . . . . . . . 9 ⊢ ((1 ∈ ℝ+ ∧ 𝑥 = (◡𝐷 “ (0[,)1))) → ∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
24	20, 23	mpan 689	. . . . . . . 8 ⊢ (𝑥 = (◡𝐷 “ (0[,)1)) → ∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
25	24	eximi 1833	. . . . . . 7 ⊢ (∃𝑥 𝑥 = (◡𝐷 “ (0[,)1)) → ∃𝑥∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
26	17, 18, 19, 25	4syl 19	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
27	2	exbidv 1920	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑥 𝑥 ∈ 𝐹 ↔ ∃𝑥∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎))))
28	26, 27	mpbird 257	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 𝑥 ∈ 𝐹)
29	28	adantl 481	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 𝑥 ∈ 𝐹)
30		n0 4376	. . . 4 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹)
31	29, 30	sylibr 234	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ≠ ∅)
32	1	metustid 24588	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥)
33	32	adantll 713	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥)
34		n0 4376	. . . . . . . . . 10 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑝 𝑝 ∈ 𝑋)
35	34	biimpi 216	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ → ∃𝑝 𝑝 ∈ 𝑋)
36	35	adantr 480	. . . . . . . 8 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑝 𝑝 ∈ 𝑋)
37		opelidres 6021	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑋 → (⟨𝑝, 𝑝⟩ ∈ ( I ↾ 𝑋) ↔ 𝑝 ∈ 𝑋))
38	37	ibir 268	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑋 → ⟨𝑝, 𝑝⟩ ∈ ( I ↾ 𝑋))
39	38	ne0d 4365	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑋 → ( I ↾ 𝑋) ≠ ∅)
40	39	exlimiv 1929	. . . . . . . 8 ⊢ (∃𝑝 𝑝 ∈ 𝑋 → ( I ↾ 𝑋) ≠ ∅)
41	36, 40	syl 17	. . . . . . 7 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ( I ↾ 𝑋) ≠ ∅)
42	41	adantr 480	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ≠ ∅)
43		ssn0 4427	. . . . . 6 ⊢ ((( I ↾ 𝑋) ⊆ 𝑥 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑥 ≠ ∅)
44	33, 42, 43	syl2anc 583	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ≠ ∅)
45	44	nelrdva 3727	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ¬ ∅ ∈ 𝐹)
46		df-nel 3053	. . . 4 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
47	45, 46	sylibr 234	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∅ ∉ 𝐹)
48		dfss2 3994	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∩ 𝑦) = 𝑥)
49	48	biimpi 216	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 → (𝑥 ∩ 𝑦) = 𝑥)
50	49	adantl 481	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∩ 𝑦) = 𝑥)
51		simplrl 776	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → 𝑥 ∈ 𝐹)
52	50, 51	eqeltrd 2844	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∩ 𝑦) ∈ 𝐹)
53		sseqin2 4244	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝑦) = 𝑦)
54	53	biimpi 216	. . . . . . . 8 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∩ 𝑦) = 𝑦)
55	54	adantl 481	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) = 𝑦)
56		simplrr 777	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ 𝐹)
57	55, 56	eqeltrd 2844	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) ∈ 𝐹)
58		simplr 768	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
59		simprl 770	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑥 ∈ 𝐹)
60		simprr 772	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑦 ∈ 𝐹)
61	1	metustto 24587	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
62	58, 59, 60, 61	syl3anc 1371	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
63	52, 57, 62	mpjaodan 959	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
64		ssidd 4032	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))
65		sseq1 4034	. . . . . 6 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
66	65	rspcev 3635	. . . . 5 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐹 ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
67	63, 64, 66	syl2anc 583	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
68	67	ralrimivva 3208	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
69	31, 47, 68	3jca 1128	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))
70		elfvex 6958	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
71	70	adantl 481	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V)
72	71, 71	xpexd 7786	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑋 × 𝑋) ∈ V)
73		isfbas2 23864	. . 3 ⊢ ((𝑋 × 𝑋) ∈ V → (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
74	72, 73	syl 17	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
75	16, 69, 74	mpbir2and 712	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946   ∉ wnel 3052  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ⟨cop 4654   ↦ cmpt 5249   I cid 5592   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  ‘cfv 6573  (class class class)co 7448  0cc0 11184  1c1 11185  ℝ^*cxr 11323  ℝ⁺crp 13057  [,)cico 13409  PsMetcpsmet 21371  fBascfbas 21375

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  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  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-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  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-po 5607  df-so 5608  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-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-rp 13058  df-ico 13413  df-psmet 21379  df-fbas 21384

This theorem is referenced by:  metust  24592  cfilucfil  24593  metuel  24598  psmetutop  24601  restmetu  24604  metucn  24605