Theorem metustid 24567

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 6023	. . 3 ⊢ Rel ( I ↾ 𝑋)
2	1	a1i 11	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → Rel ( I ↾ 𝑋))
3		vex 3484	. . . . . . . . . . . . . . 15 ⊢ 𝑞 ∈ V
4	3	brresi 6006	. . . . . . . . . . . . . 14 ⊢ (𝑝( I ↾ 𝑋)𝑞 ↔ (𝑝 ∈ 𝑋 ∧ 𝑝 I 𝑞))
5		df-br 5144	. . . . . . . . . . . . . 14 ⊢ (𝑝( I ↾ 𝑋)𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋))
6	3	ideq 5863	. . . . . . . . . . . . . . 15 ⊢ (𝑝 I 𝑞 ↔ 𝑝 = 𝑞)
7	6	anbi2i 623	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ 𝑋 ∧ 𝑝 I 𝑞) ↔ (𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞))
8	4, 5, 7	3bitr3i 301	. . . . . . . . . . . . 13 ⊢ (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) ↔ (𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞))
9	8	biimpi 216	. . . . . . . . . . . 12 ⊢ (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → (𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞))
10	9	ad2antlr 727	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝 ∈ 𝑋 ∧ 𝑝 = 𝑞))
11	10	simprd 495	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝 = 𝑞)
12		df-ov 7434	. . . . . . . . . . 11 ⊢ (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑝⟩)
13		opeq2 4874	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑞 → ⟨𝑝, 𝑝⟩ = ⟨𝑝, 𝑞⟩)
14	13	fveq2d 6910	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝐷‘⟨𝑝, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
15	12, 14	eqtrid 2789	. . . . . . . . . 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 775	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
18	10	simpld 494	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝 ∈ 𝑋)
19		psmet0 24318	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑝𝐷𝑝) = 0)
20	17, 18, 19	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝐷𝑝) = 0)
21	16, 20	eqtr3d 2779	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) = 0)
22		0xr 11308	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
23		rpxr 13044	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℝ+ → 𝑎 ∈ ℝ*)
24		rpgt0 13047	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℝ+ → 0 < 𝑎)
25		lbico1 13441	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ* ∧ 𝑎 ∈ ℝ* ∧ 0 < 𝑎) → 0 ∈ (0[,)𝑎))
26	22, 23, 24, 25	mp3an2i 1468	. . . . . . . . 9 ⊢ (𝑎 ∈ ℝ+ → 0 ∈ (0[,)𝑎))
27	26	adantl 481	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 0 ∈ (0[,)𝑎))
28	21, 27	eqeltrd 2841	. . . . . . 7 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
29		psmetf 24316	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
30	29	ffund 6740	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → Fun 𝐷)
31	30	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → Fun 𝐷)
32	11, 18	eqeltrrd 2842	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑞 ∈ 𝑋)
33	18, 32	opelxpd 5724	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
34	29	fdmd 6746	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
35	34	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
36	33, 35	eleqtrrd 2844	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
37		fvimacnv 7073	. . . . . . . 8 ⊢ ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
38	31, 36, 37	syl2anc 584	. . . . . . 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 232	. . . . . 6 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎)))
40	39	adantr 480	. . . . 5 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎)))
41		simpr 484	. . . . 5 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
42	40, 41	eleqtrrd 2844	. . . 4 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
43		simplr 769	. . . . 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 24563	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
46	45	ad2antrr 726	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
47	43, 46	mpbid 232	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
48	42, 47	r19.29a 3162	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
49	48	ex 412	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → ⟨𝑝, 𝑞⟩ ∈ 𝐴))
50	2, 49	relssdv 5798	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951  ⟨cop 4632   class class class wbr 5143   ↦ cmpt 5225   I cid 5577   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688  Rel wrel 5690  Fun wfun 6555  ‘cfv 6561  (class class class)co 7431  0cc0 11155  ℝ^*cxr 11294   < clt 11295  ℝ⁺crp 13034  [,)cico 13389  PsMetcpsmet 21348

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-addrcl 11216  ax-rnegex 11226  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-rp 13035  df-ico 13393  df-psmet 21356

This theorem is referenced by:  metustfbas  24570  metust  24571