Theorem metustsym 24459

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 24455	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
3		cnvss 5819	. . . 4 ⊢ (𝐴 ⊆ (𝑋 × 𝑋) → ◡𝐴 ⊆ ◡(𝑋 × 𝑋))
4		cnvxp 6110	. . . 4 ⊢ ◡(𝑋 × 𝑋) = (𝑋 × 𝑋)
5	3, 4	sseqtrdi 3978	. . 3 ⊢ (𝐴 ⊆ (𝑋 × 𝑋) → ◡𝐴 ⊆ (𝑋 × 𝑋))
6	2, 5	syl 17	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 ⊆ (𝑋 × 𝑋))
7		simp-4l 782	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋))
8		simpr1r 1232	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) → 𝑞 ∈ 𝑋)
9	8	3anassrs 1361	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑞 ∈ 𝑋)
10		simpr1l 1231	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) → 𝑝 ∈ 𝑋)
11	10	3anassrs 1361	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑝 ∈ 𝑋)
12		psmetsym 24214	. . . . . . . . . 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 7356	. . . . . . . . 9 ⊢ (𝑞𝐷𝑝) = (𝐷‘⟨𝑞, 𝑝⟩)
15		df-ov 7356	. . . . . . . . 9 ⊢ (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
16	13, 14, 15	3eqtr3g 2787	. . . . . . . 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 2813	. . . . . . 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 24210	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
19		ffun 6659	. . . . . . . . 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 775	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
22	21	ancomd 461	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋))
23		opelxpi 5660	. . . . . . . . . 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 6665	. . . . . . . . . 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 2831	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑞, 𝑝⟩ ∈ dom 𝐷)
28		fvimacnv 6991	. . . . . . . 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 5660	. . . . . . . . . 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 2831	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
33		fvimacnv 6991	. . . . . . . 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 309	. . . . . 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 484	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
37	36	eleq2d 2814	. . . . . 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 2814	. . . . . 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 311	. . . . 5 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴))
40		eqid 2729	. . . . . . . . 9 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
41	40	elrnmpt 5904	. . . . . . . 8 ⊢ (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) → (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
42	41	ibi 267	. . . . . . 7 ⊢ (𝐴 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
43	42, 1	eleq2s 2846	. . . . . 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 3137	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴))
46		df-br 5096	. . . . 5 ⊢ (𝑝◡𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ◡𝐴)
47		vex 3442	. . . . . 6 ⊢ 𝑝 ∈ V
48		vex 3442	. . . . . 6 ⊢ 𝑞 ∈ V
49	47, 48	opelcnv 5828	. . . . 5 ⊢ (⟨𝑝, 𝑞⟩ ∈ ◡𝐴 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
50	46, 49	bitri 275	. . . 4 ⊢ (𝑝◡𝐴𝑞 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
51		df-br 5096	. . . 4 ⊢ (𝑝𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴)
52	45, 50, 51	3bitr4g 314	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → (𝑝◡𝐴𝑞 ↔ 𝑝𝐴𝑞))
53	52	3impb 1114	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → (𝑝◡𝐴𝑞 ↔ 𝑝𝐴𝑞))
54	6, 2, 53	eqbrrdva 5816	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3905  ⟨cop 4585   class class class wbr 5095   ↦ cmpt 5176   × cxp 5621  ^◡ccnv 5622  dom cdm 5623  ran crn 5624   “ cima 5626  Fun wfun 6480  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  0cc0 11028  ℝ^*cxr 11167  ℝ⁺crp 12911  [,)cico 13268  PsMetcpsmet 21263

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-po 5531  df-so 5532  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-xadd 13033  df-psmet 21271

This theorem is referenced by:  metust  24462