Theorem metustsym 23911

Description: Elements of the filter base generated by the metric 𝐷 are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

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

Assertion

Ref	Expression
metustsym	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 = 𝐴)

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

Proof of Theorem metustsym

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metust.1	. . . 4 ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
2	1	metustss 23907	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
3		cnvss 5828	. . . 4 ⊢ (𝐴 ⊆ (𝑋 × 𝑋) → ◡𝐴 ⊆ ◡(𝑋 × 𝑋))
4		cnvxp 6109	. . . 4 ⊢ ◡(𝑋 × 𝑋) = (𝑋 × 𝑋)
5	3, 4	sseqtrdi 3994	. . 3 ⊢ (𝐴 ⊆ (𝑋 × 𝑋) → ◡𝐴 ⊆ (𝑋 × 𝑋))
6	2, 5	syl 17	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 ⊆ (𝑋 × 𝑋))
7		simp-4l 781	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋))
8		simpr1r 1231	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) → 𝑞 ∈ 𝑋)
9	8	3anassrs 1360	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑞 ∈ 𝑋)
10		simpr1l 1230	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) → 𝑝 ∈ 𝑋)
11	10	3anassrs 1360	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑝 ∈ 𝑋)
12		psmetsym 23663	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋) → (𝑞𝐷𝑝) = (𝑝𝐷𝑞))
13	7, 9, 11, 12	syl3anc 1371	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑞𝐷𝑝) = (𝑝𝐷𝑞))
14		df-ov 7360	. . . . . . . . 9 ⊢ (𝑞𝐷𝑝) = (𝐷‘⟨𝑞, 𝑝⟩)
15		df-ov 7360	. . . . . . . . 9 ⊢ (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
16	13, 14, 15	3eqtr3g 2799	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝐷‘⟨𝑞, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
17	16	eleq1d 2822	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ((𝐷‘⟨𝑞, 𝑝⟩) ∈ (0[,)𝑎) ↔ (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎)))
18		psmetf 23659	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
19		ffun 6671	. . . . . . . . 9 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → Fun 𝐷)
20	7, 18, 19	3syl 18	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → Fun 𝐷)
21		simpllr 774	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
22	21	ancomd 462	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋))
23		opelxpi 5670	. . . . . . . . . 10 ⊢ ((𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋) → ⟨𝑞, 𝑝⟩ ∈ (𝑋 × 𝑋))
24	22, 23	syl 17	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑞, 𝑝⟩ ∈ (𝑋 × 𝑋))
25		fdm 6677	. . . . . . . . . 10 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
26	7, 18, 25	3syl 18	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → dom 𝐷 = (𝑋 × 𝑋))
27	24, 26	eleqtrrd 2841	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑞, 𝑝⟩ ∈ dom 𝐷)
28		fvimacnv 7003	. . . . . . . 8 ⊢ ((Fun 𝐷 ∧ ⟨𝑞, 𝑝⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑞, 𝑝⟩) ∈ (0[,)𝑎) ↔ ⟨𝑞, 𝑝⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
29	20, 27, 28	syl2anc 584	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ((𝐷‘⟨𝑞, 𝑝⟩) ∈ (0[,)𝑎) ↔ ⟨𝑞, 𝑝⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
30		opelxpi 5670	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
31	21, 30	syl 17	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
32	31, 26	eleqtrrd 2841	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
33		fvimacnv 7003	. . . . . . . 8 ⊢ ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
34	20, 32, 33	syl2anc 584	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
35	17, 29, 34	3bitr3d 308	. . . . . 6 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (⟨𝑞, 𝑝⟩ ∈ (◡𝐷 “ (0[,)𝑎)) ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
36		simpr 485	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
37	36	eleq2d 2823	. . . . . 6 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑞, 𝑝⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
38	36	eleq2d 2823	. . . . . 6 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (⟨𝑝, 𝑞⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
39	35, 37, 38	3bitr4d 310	. . . . 5 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴))
40		eqid 2736	. . . . . . . . 9 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
41	40	elrnmpt 5911	. . . . . . . 8 ⊢ (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) → (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
42	41	ibi 266	. . . . . . 7 ⊢ (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
43	42, 1	eleq2s 2856	. . . . . 6 ⊢ (𝐴 ∈ 𝐹 → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
44	43	ad2antlr 725	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
45	39, 44	r19.29a 3159	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴))
46		df-br 5106	. . . . 5 ⊢ (𝑝◡𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ◡𝐴)
47		vex 3449	. . . . . 6 ⊢ 𝑝 ∈ V
48		vex 3449	. . . . . 6 ⊢ 𝑞 ∈ V
49	47, 48	opelcnv 5837	. . . . 5 ⊢ (⟨𝑝, 𝑞⟩ ∈ ◡𝐴 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
50	46, 49	bitri 274	. . . 4 ⊢ (𝑝◡𝐴𝑞 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
51		df-br 5106	. . . 4 ⊢ (𝑝𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴)
52	45, 50, 51	3bitr4g 313	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → (𝑝◡𝐴𝑞 ↔ 𝑝𝐴𝑞))
53	52	3impb 1115	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → (𝑝◡𝐴𝑞 ↔ 𝑝𝐴𝑞))
54	6, 2, 53	eqbrrdva 5825	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 = 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3073   ⊆ wss 3910  ⟨cop 4592   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631  ^◡ccnv 5632  dom cdm 5633  ran crn 5634   “ cima 5636  Fun wfun 6490  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  0cc0 11051  ℝ^*cxr 11188  ℝ⁺crp 12915  [,)cico 13266  PsMetcpsmet 20780

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-xadd 13034  df-psmet 20788

This theorem is referenced by:  metust  23914