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Theorem metustfbas 22172

Description: The filter base generated by a metric 𝐷. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

**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 22165	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ⁺ 𝑥 = (^◡𝐷 “ (0[,)𝑎))))
3		simpr 476	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (^◡𝐷 “ (0[,)𝑎))) → 𝑥 = (^◡𝐷 “ (0[,)𝑎)))
4		cnvimass 5404	. . . . . . . . . 10 ⊢ (^◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
5		psmetf 21921	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
6		fdm 5964	. . . . . . . . . . . 12 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → dom 𝐷 = (𝑋 × 𝑋))
7	5, 6	syl 17	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
8	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (^◡𝐷 “ (0[,)𝑎))) → dom 𝐷 = (𝑋 × 𝑋))
9	4, 8	syl5sseq 3616	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (^◡𝐷 “ (0[,)𝑎))) → (^◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
10	3, 9	eqsstrd 3602	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (^◡𝐷 “ (0[,)𝑎))) → 𝑥 ⊆ (𝑋 × 𝑋))
11	10	ex 449	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 = (^◡𝐷 “ (0[,)𝑎)) → 𝑥 ⊆ (𝑋 × 𝑋)))
12	11	rexlimdvw 3016	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ⁺ 𝑥 = (^◡𝐷 “ (0[,)𝑎)) → 𝑥 ⊆ (𝑋 × 𝑋)))
13	2, 12	sylbid 229	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ (𝑋 × 𝑋)))
14	13	ralrimiv 2948	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝐹 𝑥 ⊆ (𝑋 × 𝑋))
15		pwssb 4548	. . . 4 ⊢ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑥 ∈ 𝐹 𝑥 ⊆ (𝑋 × 𝑋))
16	14, 15	sylibr 223	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
17	16	adantl 481	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
18		cnvexg 7005	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ^◡𝐷 ∈ V)
19		imaexg 6995	. . . . . . 7 ⊢ (^◡𝐷 ∈ V → (^◡𝐷 “ (0[,)1)) ∈ V)
20		elisset 3188	. . . . . . 7 ⊢ ((^◡𝐷 “ (0[,)1)) ∈ V → ∃𝑥 𝑥 = (^◡𝐷 “ (0[,)1)))
21		1rp 11712	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
22		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑎 = 1 → (0[,)𝑎) = (0[,)1))
23	22	imaeq2d 5385	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → (^◡𝐷 “ (0[,)𝑎)) = (^◡𝐷 “ (0[,)1)))
24	23	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑎 = 1 → (𝑥 = (^◡𝐷 “ (0[,)𝑎)) ↔ 𝑥 = (^◡𝐷 “ (0[,)1))))
25	24	rspcev 3282	. . . . . . . . 9 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑥 = (^◡𝐷 “ (0[,)1))) → ∃𝑎 ∈ ℝ⁺ 𝑥 = (^◡𝐷 “ (0[,)𝑎)))
26	21, 25	mpan 702	. . . . . . . 8 ⊢ (𝑥 = (^◡𝐷 “ (0[,)1)) → ∃𝑎 ∈ ℝ⁺ 𝑥 = (^◡𝐷 “ (0[,)𝑎)))
27	26	eximi 1752	. . . . . . 7 ⊢ (∃𝑥 𝑥 = (^◡𝐷 “ (0[,)1)) → ∃𝑥∃𝑎 ∈ ℝ⁺ 𝑥 = (^◡𝐷 “ (0[,)𝑎)))
28	18, 19, 20, 27	4syl 19	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥∃𝑎 ∈ ℝ⁺ 𝑥 = (^◡𝐷 “ (0[,)𝑎)))
29	2	exbidv 1837	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑥 𝑥 ∈ 𝐹 ↔ ∃𝑥∃𝑎 ∈ ℝ⁺ 𝑥 = (^◡𝐷 “ (0[,)𝑎))))
30	28, 29	mpbird 246	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 𝑥 ∈ 𝐹)
31	30	adantl 481	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 𝑥 ∈ 𝐹)
32		n0 3890	. . . 4 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹)
33	31, 32	sylibr 223	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ≠ ∅)
34	1	metustid 22169	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥)
35	34	adantll 746	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥)
36		n0 3890	. . . . . . . . . 10 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑝 𝑝 ∈ 𝑋)
37	36	biimpi 205	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ → ∃𝑝 𝑝 ∈ 𝑋)
38	37	adantr 480	. . . . . . . 8 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑝 𝑝 ∈ 𝑋)
39		opelresi 5328	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑋 → (⟨𝑝, 𝑝⟩ ∈ ( I ↾ 𝑋) ↔ 𝑝 ∈ 𝑋))
40	39	ibir 256	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑋 → ⟨𝑝, 𝑝⟩ ∈ ( I ↾ 𝑋))
41		ne0i 3880	. . . . . . . . . 10 ⊢ (⟨𝑝, 𝑝⟩ ∈ ( I ↾ 𝑋) → ( I ↾ 𝑋) ≠ ∅)
42	40, 41	syl 17	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑋 → ( I ↾ 𝑋) ≠ ∅)
43	42	exlimiv 1845	. . . . . . . 8 ⊢ (∃𝑝 𝑝 ∈ 𝑋 → ( I ↾ 𝑋) ≠ ∅)
44	38, 43	syl 17	. . . . . . 7 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ( I ↾ 𝑋) ≠ ∅)
45	44	adantr 480	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ≠ ∅)
46		ssn0 3928	. . . . . 6 ⊢ ((( I ↾ 𝑋) ⊆ 𝑥 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑥 ≠ ∅)
47	35, 45, 46	syl2anc 691	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ≠ ∅)
48	47	nelrdva 3384	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ¬ ∅ ∈ 𝐹)
49		df-nel 2783	. . . 4 ⊢ (∅ ∉ 𝐹 ↔ ¬ ∅ ∈ 𝐹)
50	48, 49	sylibr 223	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∅ ∉ 𝐹)
51		df-ss 3554	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∩ 𝑦) = 𝑥)
52	51	biimpi 205	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝑦 → (𝑥 ∩ 𝑦) = 𝑥)
53	52	adantl 481	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∩ 𝑦) = 𝑥)
54		simplrl 796	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → 𝑥 ∈ 𝐹)
55	53, 54	eqeltrd 2688	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∩ 𝑦) ∈ 𝐹)
56		sseqin2 3779	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝑦) = 𝑦)
57	56	biimpi 205	. . . . . . . 8 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∩ 𝑦) = 𝑦)
58	57	adantl 481	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) = 𝑦)
59		simplrr 797	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ 𝐹)
60	58, 59	eqeltrd 2688	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) ∈ 𝐹)
61		simplr 788	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝐷 ∈ (PsMet‘𝑋))
62		simprl 790	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑥 ∈ 𝐹)
63		simprr 792	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑦 ∈ 𝐹)
64	1	metustto 22168	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
65	61, 62, 63, 64	syl3anc 1318	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
66	55, 60, 65	mpjaodan 823	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
67		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
68	67	inex1 4727	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ∈ V
69	68	pwid 4122	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦)
70	69	a1i 11	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝒫 (𝑥 ∩ 𝑦))
71	70	elpwid 4118	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))
72		sseq1 3589	. . . . . 6 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
73	72	rspcev 3282	. . . . 5 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐹 ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
74	66, 71, 73	syl2anc 691	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
75	74	ralrimivva 2954	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))
76	33, 50, 75	3jca 1235	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))
77		elfvex 6131	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
78	77	adantl 481	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V)
79		xpexg 6858	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
80	78, 78, 79	syl2anc 691	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑋 × 𝑋) ∈ V)
81		isfbas2 21449	. . 3 ⊢ ((𝑋 × 𝑋) ∈ V → (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
82	80, 81	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)))))
83	17, 76, 82	mpbir2and 959	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ⟨cop 4131 ↦ cmpt 4643 I cid 4948 × cxp 5036 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ℝ^*cxr 9952 ℝ⁺crp 11708 [,)cico 12048 PsMetcpsmet 19551 fBascfbas 19555

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-rp 11709 df-ico 12052 df-psmet 19559 df-fbas 19564

This theorem is referenced by: metust 22173 cfilucfil 22174 metuel 22179 psmetutop 22182 restmetu 22185 metucn 22186

W3C validator