Theorem metustsym 23453

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 23449	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
3		cnvss 5741	. . . 4 ⊢ (𝐴 ⊆ (𝑋 × 𝑋) → ◡𝐴 ⊆ ◡(𝑋 × 𝑋))
4		cnvxp 6020	. . . 4 ⊢ ◡(𝑋 × 𝑋) = (𝑋 × 𝑋)
5	3, 4	sseqtrdi 3951	. . 3 ⊢ (𝐴 ⊆ (𝑋 × 𝑋) → ◡𝐴 ⊆ (𝑋 × 𝑋))
6	2, 5	syl 17	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 ⊆ (𝑋 × 𝑋))
7		simp-4l 783	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋))
8		simpr1r 1233	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) → 𝑞 ∈ 𝑋)
9	8	3anassrs 1362	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑞 ∈ 𝑋)
10		simpr1l 1232	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) → 𝑝 ∈ 𝑋)
11	10	3anassrs 1362	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑝 ∈ 𝑋)
12		psmetsym 23208	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋) → (𝑞𝐷𝑝) = (𝑝𝐷𝑞))
13	7, 9, 11, 12	syl3anc 1373	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑞𝐷𝑝) = (𝑝𝐷𝑞))
14		df-ov 7216	. . . . . . . . 9 ⊢ (𝑞𝐷𝑝) = (𝐷‘⟨𝑞, 𝑝⟩)
15		df-ov 7216	. . . . . . . . 9 ⊢ (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
16	13, 14, 15	3eqtr3g 2801	. . . . . . . 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 23204	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
19		ffun 6548	. . . . . . . . 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 776	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
22	21	ancomd 465	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋))
23		opelxpi 5588	. . . . . . . . . 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 6554	. . . . . . . . . 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 6873	. . . . . . . 8 ⊢ ((Fun 𝐷 ∧ ⟨𝑞, 𝑝⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑞, 𝑝⟩) ∈ (0[,)𝑎) ↔ ⟨𝑞, 𝑝⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
29	20, 27, 28	syl2anc 587	. . . . . . 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 5588	. . . . . . . . . 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 6873	. . . . . . . 8 ⊢ ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
34	20, 32, 33	syl2anc 587	. . . . . . 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 312	. . . . . 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 488	. . . . . . 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 314	. . . . 5 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴))
40		eqid 2737	. . . . . . . . 9 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
41	40	elrnmpt 5825	. . . . . . . 8 ⊢ (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) → (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
42	41	ibi 270	. . . . . . 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 727	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
45	39, 44	r19.29a 3208	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴))
46		df-br 5054	. . . . 5 ⊢ (𝑝◡𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ◡𝐴)
47		vex 3412	. . . . . 6 ⊢ 𝑝 ∈ V
48		vex 3412	. . . . . 6 ⊢ 𝑞 ∈ V
49	47, 48	opelcnv 5750	. . . . 5 ⊢ (⟨𝑝, 𝑞⟩ ∈ ◡𝐴 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
50	46, 49	bitri 278	. . . 4 ⊢ (𝑝◡𝐴𝑞 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
51		df-br 5054	. . . 4 ⊢ (𝑝𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴)
52	45, 50, 51	3bitr4g 317	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → (𝑝◡𝐴𝑞 ↔ 𝑝𝐴𝑞))
53	52	3impb 1117	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → (𝑝◡𝐴𝑞 ↔ 𝑝𝐴𝑞))
54	6, 2, 53	eqbrrdva 5738	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 = 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∃wrex 3062   ⊆ wss 3866  ⟨cop 4547   class class class wbr 5053   ↦ cmpt 5135   × cxp 5549  ^◡ccnv 5550  dom cdm 5551  ran crn 5552   “ cima 5554  Fun wfun 6374  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  0cc0 10729  ℝ^*cxr 10866  ℝ⁺crp 12586  [,)cico 12937  PsMetcpsmet 20347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-po 5468  df-so 5469  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-xadd 12705  df-psmet 20355

This theorem is referenced by:  metust  23456