Theorem metustid 23161

Description: The identity diagonal is included in all elements of the filter base generated by the metric 𝐷. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Hypothesis

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

Assertion

Ref	Expression
metustid	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)

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

Proof of Theorem metustid

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

Step	Hyp	Ref	Expression
1		relres 5847	. . 3 ⊢ Rel ( I ↾ 𝑋)
2	1	a1i 11	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → Rel ( I ↾ 𝑋))
3		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑞 ∈ V
4	3	brresi 5827	. . . . . . . . . . . . . 14 ⊢ (𝑝( I ↾ 𝑋)𝑞 ↔ (𝑝 ∈ 𝑋 ∧ 𝑝 I 𝑞))
5		df-br 5031	. . . . . . . . . . . . . 14 ⊢ (𝑝( I ↾ 𝑋)𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋))
6	3	ideq 5687	. . . . . . . . . . . . . . 15 ⊢ (𝑝 I 𝑞 ↔ 𝑝 = 𝑞)
7	6	anbi2i 625	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ 𝑋 ∧ 𝑝 I 𝑞) ↔ (𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞))
8	4, 5, 7	3bitr3i 304	. . . . . . . . . . . . 13 ⊢ (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) ↔ (𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞))
9	8	biimpi 219	. . . . . . . . . . . 12 ⊢ (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → (𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞))
10	9	ad2antlr 726	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞))
11	10	simprd 499	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝 = 𝑞)
12		df-ov 7138	. . . . . . . . . . 11 ⊢ (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑝⟩)
13		opeq2 4765	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑞 → ⟨𝑝, 𝑝⟩ = ⟨𝑝, 𝑞⟩)
14	13	fveq2d 6649	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝐷‘⟨𝑝, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
15	12, 14	syl5eq 2845	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑞⟩))
16	11, 15	syl 17	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑞⟩))
17		simplll 774	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
18	10	simpld 498	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝 ∈ 𝑋)
19		psmet0 22915	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑝𝐷𝑝) = 0)
20	17, 18, 19	syl2anc 587	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝐷𝑝) = 0)
21	16, 20	eqtr3d 2835	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) = 0)
22		0xr 10677	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
23		rpxr 12386	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℝ+ → 𝑎 ∈ ℝ*)
24		rpgt0 12389	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℝ+ → 0 < 𝑎)
25		lbico1 12779	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ* ∧ 𝑎 ∈ ℝ* ∧ 0 < 𝑎) → 0 ∈ (0[,)𝑎))
26	22, 23, 24, 25	mp3an2i 1463	. . . . . . . . 9 ⊢ (𝑎 ∈ ℝ+ → 0 ∈ (0[,)𝑎))
27	26	adantl 485	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 0 ∈ (0[,)𝑎))
28	21, 27	eqeltrd 2890	. . . . . . 7 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
29		psmetf 22913	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
30	29	ffund 6491	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → Fun 𝐷)
31	30	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → Fun 𝐷)
32	11, 18	eqeltrrd 2891	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑞 ∈ 𝑋)
33	18, 32	opelxpd 5557	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
34	29	fdmd 6497	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
35	34	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
36	33, 35	eleqtrrd 2893	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
37		fvimacnv 6800	. . . . . . . 8 ⊢ ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
38	31, 36, 37	syl2anc 587	. . . . . . 7 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
39	28, 38	mpbid 235	. . . . . 6 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎)))
40	39	adantr 484	. . . . 5 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎)))
41		simpr 488	. . . . 5 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
42	40, 41	eleqtrrd 2893	. . . 4 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
43		simplr 768	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → 𝐴 ∈ 𝐹)
44		metust.1	. . . . . . 7 ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
45	44	metustel 23157	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
46	45	ad2antrr 725	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
47	43, 46	mpbid 235	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
48	42, 47	r19.29a 3248	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
49	48	ex 416	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → ⟨𝑝, 𝑞⟩ ∈ 𝐴))
50	2, 49	relssdv 5625	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ⊆ wss 3881  ⟨cop 4531   class class class wbr 5030   ↦ cmpt 5110   I cid 5424   × cxp 5517  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   ↾ cres 5521   “ cima 5522  Rel wrel 5524  Fun wfun 6318  ‘cfv 6324  (class class class)co 7135  0cc0 10526  ℝ^*cxr 10663   < clt 10664  ℝ⁺crp 12377  [,)cico 12728  PsMetcpsmet 20075

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-addrcl 10587  ax-rnegex 10597  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-rp 12378  df-ico 12732  df-psmet 20083

This theorem is referenced by:  metustfbas  23164  metust  23165