Theorem metustfbas 24522

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 24515	. . . . . 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 6048	. . . . . . . . . 10 ⊢ (◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
5		psmetf 24271	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	fdmd 6679	. . . . . . . . . . 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 3965	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
9	3, 8	eqsstrd 3957	. . . . . . . 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 3144	. . . . . 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 3129	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝐹 𝑥 ⊆ (𝑋 × 𝑋))
14		pwssb 5044	. . . 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 7875	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
18		imaexg 7864	. . . . . . 7 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)1)) ∈ V)
19		elisset 2819	. . . . . . 7 ⊢ ((◡𝐷 “ (0[,)1)) ∈ V → ∃𝑥 𝑥 = (◡𝐷 “ (0[,)1)))
20		1rp 12946	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
21		oveq2 7375	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → (0[,)𝑎) = (0[,)1))
22	21	imaeq2d 6026	. . . . . . . . . 10 ⊢ (𝑎 = 1 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)1)))
23	22	rspceeqv 3588	. . . . . . . . 9 ⊢ ((1 ∈ ℝ+ ∧ 𝑥 = (◡𝐷 “ (0[,)1))) → ∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
24	20, 23	mpan 691	. . . . . . . 8 ⊢ (𝑥 = (◡𝐷 “ (0[,)1)) → ∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
25	24	eximi 1837	. . . . . . 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 1923	. . . . . 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 4294	. . . 4 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹)
31	29, 30	sylibr 234	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ≠ ∅)
32	1	metustid 24519	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥)
33	32	adantll 715	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥)
34		n0 4294	. . . . . . . . . 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 5957	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑋 → (⟨𝑝, 𝑝⟩ ∈ ( I ↾ 𝑋) ↔ 𝑝 ∈ 𝑋))
38	37	ibir 268	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑋 → ⟨𝑝, 𝑝⟩ ∈ ( I ↾ 𝑋))
39	38	ne0d 4283	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑋 → ( I ↾ 𝑋) ≠ ∅)
40	39	exlimiv 1932	. . . . . . . 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 4345	. . . . . 6 ⊢ ((( I ↾ 𝑋) ⊆ 𝑥 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑥 ≠ ∅)
44	33, 42, 43	syl2anc 585	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ≠ ∅)
45	44	nelrdva 3652	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ¬ ∅ ∈ 𝐹)
46		df-nel 3038	. . . 4 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
47	45, 46	sylibr 234	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∅ ∉ 𝐹)
48		dfss2 3908	. . . . . . . . 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 777	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → 𝑥 ∈ 𝐹)
52	50, 51	eqeltrd 2837	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∩ 𝑦) ∈ 𝐹)
53		sseqin2 4164	. . . . . . . . 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 778	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ 𝐹)
57	55, 56	eqeltrd 2837	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) ∈ 𝐹)
58		simplr 769	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
59		simprl 771	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑥 ∈ 𝐹)
60		simprr 773	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑦 ∈ 𝐹)
61	1	metustto 24518	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
62	58, 59, 60, 61	syl3anc 1374	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
63	52, 57, 62	mpjaodan 961	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
64		ssidd 3946	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))
65		sseq1 3948	. . . . . 6 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
66	65	rspcev 3565	. . . . 5 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐹 ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
67	63, 64, 66	syl2anc 585	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
68	67	ralrimivva 3181	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
69	31, 47, 68	3jca 1129	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))
70		elfvex 6876	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
71	70	adantl 481	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V)
72	71, 71	xpexd 7705	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑋 × 𝑋) ∈ V)
73		isfbas2 23800	. . 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 714	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933   ∉ wnel 3037  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ⟨cop 4574   ↦ cmpt 5167   I cid 5525   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634  ‘cfv 6499  (class class class)co 7367  0cc0 11038  1c1 11039  ℝ^*cxr 11178  ℝ⁺crp 12942  [,)cico 13300  PsMetcpsmet 21336  fBascfbas 21340

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 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  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 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-rp 12943  df-ico 13304  df-psmet 21344  df-fbas 21349

This theorem is referenced by:  metust  24523  cfilucfil  24524  metuel  24529  psmetutop  24532  restmetu  24535  metucn  24536