Theorem metustid 24449

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 5979	. . 3 ⊢ Rel ( I ↾ 𝑋)
2	1	a1i 11	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → Rel ( I ↾ 𝑋))
3		vex 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑞 ∈ V
4	3	brresi 5962	. . . . . . . . . . . . . 14 ⊢ (𝑝( I ↾ 𝑋)𝑞 ↔ (𝑝 ∈ 𝑋 ∧ 𝑝 I 𝑞))
5		df-br 5111	. . . . . . . . . . . . . 14 ⊢ (𝑝( I ↾ 𝑋)𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋))
6	3	ideq 5819	. . . . . . . . . . . . . . 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 7393	. . . . . . . . . . 11 ⊢ (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑝⟩)
13		opeq2 4841	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑞 → ⟨𝑝, 𝑝⟩ = ⟨𝑝, 𝑞⟩)
14	13	fveq2d 6865	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝐷‘⟨𝑝, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
15	12, 14	eqtrid 2777	. . . . . . . . . 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 494	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝 ∈ 𝑋)
19		psmet0 24203	. . . . . . . . . 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 2767	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) = 0)
22		0xr 11228	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
23		rpxr 12968	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℝ+ → 𝑎 ∈ ℝ*)
24		rpgt0 12971	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℝ+ → 0 < 𝑎)
25		lbico1 13368	. . . . . . . . . 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 2829	. . . . . . 7 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
29		psmetf 24201	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
30	29	ffund 6695	. . . . . . . . 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 2830	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑞 ∈ 𝑋)
33	18, 32	opelxpd 5680	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
34	29	fdmd 6701	. . . . . . . . . 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 2832	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
37		fvimacnv 7028	. . . . . . . 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 2832	. . . 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 24445	. . . . . 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 3142	. . 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 5754	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3054   ⊆ wss 3917  ⟨cop 4598   class class class wbr 5110   ↦ cmpt 5191   I cid 5535   × cxp 5639  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   ↾ cres 5643   “ cima 5644  Rel wrel 5646  Fun wfun 6508  ‘cfv 6514  (class class class)co 7390  0cc0 11075  ℝ^*cxr 11214   < clt 11215  ℝ⁺crp 12958  [,)cico 13315  PsMetcpsmet 21255

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-addrcl 11136  ax-rnegex 11146  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-rp 12959  df-ico 13319  df-psmet 21263

This theorem is referenced by:  metustfbas  24452  metust  24453