Theorem metustfbas 24693

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 24686	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎))))
3		simpr 489	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → 𝑥 = (◡𝐷 “ (0[,)𝑎)))
4		cnvimass 6084	. . . . . . . . . 10 ⊢ (◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
5		psmetf 24442	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	fdmd 6716	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
7	6	adantr 485	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → dom 𝐷 = (𝑋 × 𝑋))
8	4, 7	sseqtrid 3978	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
9	3, 8	eqsstrd 3970	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → 𝑥 ⊆ (𝑋 × 𝑋))
10	9	ex 417	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 = (◡𝐷 “ (0[,)𝑎)) → 𝑥 ⊆ (𝑋 × 𝑋)))
11	10	rexlimdvw 3169	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)) → 𝑥 ⊆ (𝑋 × 𝑋)))
12	2, 11	sylbid 243	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ (𝑋 × 𝑋)))
13	12	ralrimiv 3154	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝐹 𝑥 ⊆ (𝑋 × 𝑋))
14		pwssb 5066	. . . 4 ⊢ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑥 ∈ 𝐹 𝑥 ⊆ (𝑋 × 𝑋))
15	13, 14	sylibr 237	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
16	15	adantl 486	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
17		cnvexg 7920	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
18		imaexg 7909	. . . . . . 7 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)1)) ∈ V)
19		elisset 2843	. . . . . . 7 ⊢ ((◡𝐷 “ (0[,)1)) ∈ V → ∃𝑥 𝑥 = (◡𝐷 “ (0[,)1)))
20		1rp 13019	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
21		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → (0[,)𝑎) = (0[,)1))
22	21	imaeq2d 6062	. . . . . . . . . 10 ⊢ (𝑎 = 1 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)1)))
23	22	rspceeqv 3603	. . . . . . . . 9 ⊢ ((1 ∈ ℝ+ ∧ 𝑥 = (◡𝐷 “ (0[,)1))) → ∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
24	20, 23	mpan 702	. . . . . . . 8 ⊢ (𝑥 = (◡𝐷 “ (0[,)1)) → ∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
25	24	eximi 1863	. . . . . . 7 ⊢ (∃𝑥 𝑥 = (◡𝐷 “ (0[,)1)) → ∃𝑥∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
26	17, 18, 19, 25	4syl 20	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))
27	2	exbidv 1949	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑥 𝑥 ∈ 𝐹 ↔ ∃𝑥∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎))))
28	26, 27	mpbird 260	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 𝑥 ∈ 𝐹)
29	28	adantl 486	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 𝑥 ∈ 𝐹)
30		n0 4306	. . . 4 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹)
31	29, 30	sylibr 237	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ≠ ∅)
32	1	metustid 24690	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥)
33	32	adantll 726	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥)
34		n0 4306	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑝 𝑝 ∈ 𝑋)
35	34	birani 508	. . . . . . . 8 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑝 𝑝 ∈ 𝑋)
36		opelidres 5990	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑋 → (⟨𝑝, 𝑝⟩ ∈ ( I ↾ 𝑋) ↔ 𝑝 ∈ 𝑋))
37	36	ibir 271	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑋 → ⟨𝑝, 𝑝⟩ ∈ ( I ↾ 𝑋))
38	37	ne0d 4294	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑋 → ( I ↾ 𝑋) ≠ ∅)
39	38	exlimiv 1958	. . . . . . . 8 ⊢ (∃𝑝 𝑝 ∈ 𝑋 → ( I ↾ 𝑋) ≠ ∅)
40	35, 39	syl 18	. . . . . . 7 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ( I ↾ 𝑋) ≠ ∅)
41	40	adantr 485	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ≠ ∅)
42		ssn0 4361	. . . . . 6 ⊢ ((( I ↾ 𝑋) ⊆ 𝑥 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑥 ≠ ∅)
43	33, 41, 42	syl2anc 595	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ≠ ∅)
44	43	nelrdva 3667	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ¬ ∅ ∈ 𝐹)
45		df-nel 3063	. . . 4 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
46	44, 45	sylibr 237	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∅ ∉ 𝐹)
47		dfss2 3922	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∩ 𝑦) = 𝑥)
48	47	bilani 509	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∩ 𝑦) = 𝑥)
49		simplrl 788	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → 𝑥 ∈ 𝐹)
50	48, 49	eqeltrd 2861	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∩ 𝑦) ∈ 𝐹)
51		sseqin2 4175	. . . . . . . 8 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝑦) = 𝑦)
52	51	bilani 509	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) = 𝑦)
53		simplrr 789	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ 𝐹)
54	52, 53	eqeltrd 2861	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) ∈ 𝐹)
55		simplr 780	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
56		simprl 782	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑥 ∈ 𝐹)
57		simprr 784	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑦 ∈ 𝐹)
58	1	metustto 24689	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
59	55, 56, 57, 58	syl3anc 1396	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
60	50, 54, 59	mpjaodan 973	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
61		ssidd 3959	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))
62		sseq1 3961	. . . . . 6 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
63	62	rspcev 3580	. . . . 5 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐹 ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
64	60, 61, 63	syl2anc 595	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
65	64	ralrimivva 3206	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
66	31, 46, 65	3jca 1144	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))
67		elfvex 6916	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
68	67	adantl 486	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V)
69	68, 68	xpexd 7749	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑋 × 𝑋) ∈ V)
70		isfbas2 23971	. . 3 ⊢ ((𝑋 × 𝑋) ∈ V → (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
71	69, 70	syl 18	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
72	16, 66, 71	mpbir2and 725	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1568  ∃wex 1807   ∈ wcel 2141   ≠ wne 2956   ∉ wnel 3062  ∀wral 3077  ∃wrex 3087  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561  ⟨cop 4594   ↦ cmpt 5191   I cid 5555   × cxp 5659  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664  ‘cfv 6536  (class class class)co 7410  0cc0 11099  1c1 11100  ℝ^*cxr 11241  ℝ⁺crp 13015  [,)cico 13373  PsMetcpsmet 21485  fBascfbas 21489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-rp 13016  df-ico 13377  df-psmet 21493  df-fbas 21498

This theorem is referenced by:  metust  24694  cfilucfil  24695  metuel  24700  psmetutop  24703  restmetu  24706  metucn  24707