Theorem metustsym 24836

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 24832	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
3		cnvss 5846	. . . 4 ⊢ (𝐴 ⊆ (𝑋 × 𝑋) → ◡𝐴 ⊆ ◡(𝑋 × 𝑋))
4		cnvxp 6142	. . . 4 ⊢ ◡(𝑋 × 𝑋) = (𝑋 × 𝑋)
5	3, 4	sseqtrdi 3970	. . 3 ⊢ (𝐴 ⊆ (𝑋 × 𝑋) → ◡𝐴 ⊆ (𝑋 × 𝑋))
6	2, 5	syl 18	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 ⊆ (𝑋 × 𝑋))
7		simp-4l 795	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋))
8		simpr1r 1250	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) → 𝑞 ∈ 𝑋)
9	8	3anassrs 1381	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑞 ∈ 𝑋)
10		simpr1l 1249	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑎 ∈ ℝ+ ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) → 𝑝 ∈ 𝑋)
11	10	3anassrs 1381	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑝 ∈ 𝑋)
12		psmetsym 24591	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋) → (𝑞𝐷𝑝) = (𝑝𝐷𝑞))
13	7, 9, 11, 12	syl3anc 1398	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑞𝐷𝑝) = (𝑝𝐷𝑞))
14		df-ov 7411	. . . . . . . . 9 ⊢ (𝑞𝐷𝑝) = (𝐷‘⟨𝑞, 𝑝⟩)
15		df-ov 7411	. . . . . . . . 9 ⊢ (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
16	13, 14, 15	3eqtr3g 2818	. . . . . . . 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 2845	. . . . . . 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 24587	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
19		ffun 6700	. . . . . . . . 9 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → Fun 𝐷)
20	7, 18, 19	3syl 19	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → Fun 𝐷)
21		simpllr 788	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
22	21	ancomd 467	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋))
23		opelxpi 5684	. . . . . . . . . 10 ⊢ ((𝑞 ∈ 𝑋 ∧ 𝑝 ∈ 𝑋) → ⟨𝑞, 𝑝⟩ ∈ (𝑋 × 𝑋))
24	22, 23	syl 18	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑞, 𝑝⟩ ∈ (𝑋 × 𝑋))
25		fdm 6707	. . . . . . . . . 10 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
26	7, 18, 25	3syl 19	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → dom 𝐷 = (𝑋 × 𝑋))
27	24, 26	eleqtrrd 2863	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑞, 𝑝⟩ ∈ dom 𝐷)
28		fvimacnv 7040	. . . . . . . 8 ⊢ ((Fun 𝐷 ∧ ⟨𝑞, 𝑝⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑞, 𝑝⟩) ∈ (0[,)𝑎) ↔ ⟨𝑞, 𝑝⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
29	20, 27, 28	syl2anc 596	. . . . . . 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 5684	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
31	21, 30	syl 18	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
32	31, 26	eleqtrrd 2863	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
33		fvimacnv 7040	. . . . . . . 8 ⊢ ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
34	20, 32, 33	syl2anc 596	. . . . . . 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 490	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
37	36	eleq2d 2846	. . . . . 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 2846	. . . . . 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 2760	. . . . . . . . 9 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
41	40	elrnmpt 5936	. . . . . . . 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 2878	. . . . . 6 ⊢ (𝐴 ∈ 𝐹 → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
44	43	ad2antlr 740	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
45	39, 44	r19.29a 3170	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → (⟨𝑞, 𝑝⟩ ∈ 𝐴 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴))
46		df-br 5103	. . . . 5 ⊢ (𝑝◡𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ◡𝐴)
47		vex 3454	. . . . . 6 ⊢ 𝑝 ∈ V
48		vex 3454	. . . . . 6 ⊢ 𝑞 ∈ V
49	47, 48	opelcnv 5855	. . . . 5 ⊢ (⟨𝑝, 𝑞⟩ ∈ ◡𝐴 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
50	46, 49	bitri 278	. . . 4 ⊢ (𝑝◡𝐴𝑞 ↔ ⟨𝑞, 𝑝⟩ ∈ 𝐴)
51		df-br 5103	. . . 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 1132	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → (𝑝◡𝐴𝑞 ↔ 𝑝𝐴𝑞))
54	6, 2, 53	eqbrrdva 5843	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ◡𝐴 = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3898  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185   × cxp 5645  ^◡ccnv 5646  dom cdm 5647  ran crn 5648   “ cima 5650  Fun wfun 6521  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408  0cc0 11172  ℝ^*cxr 11314  ℝ⁺crp 13090  [,)cico 13448  PsMetcpsmet 21624

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11228  ax-resscn 11229  ax-1cn 11230  ax-icn 11231  ax-addcl 11232  ax-addrcl 11233  ax-mulcl 11234  ax-mulrcl 11235  ax-mulcom 11236  ax-addass 11237  ax-mulass 11238  ax-distr 11239  ax-i2m1 11240  ax-1ne0 11241  ax-1rid 11242  ax-rnegex 11243  ax-rrecex 11244  ax-cnre 11245  ax-pre-lttri 11246  ax-pre-lttrn 11247  ax-pre-ltadd 11248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-po 5555  df-so 5556  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-er 8695  df-map 8827  df-en 8952  df-dom 8953  df-sdom 8954  df-pnf 11317  df-mnf 11318  df-xr 11319  df-ltxr 11320  df-le 11321  df-xadd 13212  df-psmet 21632

This theorem is used by:  metust  24839