Theorem metustid 24478

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 5989	. . 3 ⊢ Rel ( I ↾ 𝑋)
2	1	a1i 11	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → Rel ( I ↾ 𝑋))
3		vex 3461	. . . . . . . . . . . . . . 15 ⊢ 𝑞 ∈ V
4	3	brresi 5972	. . . . . . . . . . . . . 14 ⊢ (𝑝( I ↾ 𝑋)𝑞 ↔ (𝑝 ∈ 𝑋 ∧ 𝑝 I 𝑞))
5		df-br 5117	. . . . . . . . . . . . . 14 ⊢ (𝑝( I ↾ 𝑋)𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋))
6	3	ideq 5829	. . . . . . . . . . . . . . 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 7402	. . . . . . . . . . 11 ⊢ (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑝⟩)
13		opeq2 4847	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑞 → ⟨𝑝, 𝑝⟩ = ⟨𝑝, 𝑞⟩)
14	13	fveq2d 6876	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝐷‘⟨𝑝, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
15	12, 14	eqtrid 2781	. . . . . . . . . 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 24232	. . . . . . . . . 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 2771	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) = 0)
22		0xr 11274	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
23		rpxr 13010	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℝ+ → 𝑎 ∈ ℝ*)
24		rpgt0 13013	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℝ+ → 0 < 𝑎)
25		lbico1 13407	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ* ∧ 𝑎 ∈ ℝ* ∧ 0 < 𝑎) → 0 ∈ (0[,)𝑎))
26	22, 23, 24, 25	mp3an2i 1467	. . . . . . . . 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 2833	. . . . . . 7 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
29		psmetf 24230	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
30	29	ffund 6706	. . . . . . . . 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 2834	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑞 ∈ 𝑋)
33	18, 32	opelxpd 5690	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
34	29	fdmd 6712	. . . . . . . . . 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 2836	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
37		fvimacnv 7039	. . . . . . . 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 2836	. . . 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 24474	. . . . . 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 3146	. . 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 5764	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃wrex 3059   ⊆ wss 3924  ⟨cop 4605   class class class wbr 5116   ↦ cmpt 5198   I cid 5544   × cxp 5649  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  Rel wrel 5656  Fun wfun 6521  ‘cfv 6527  (class class class)co 7399  0cc0 11121  ℝ^*cxr 11260   < clt 11261  ℝ⁺crp 13000  [,)cico 13355  PsMetcpsmet 21284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723  ax-cnex 11177  ax-resscn 11178  ax-1cn 11179  ax-addrcl 11182  ax-rnegex 11192  ax-cnre 11194  ax-pre-lttri 11195  ax-pre-lttrn 11196

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-po 5558  df-so 5559  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7402  df-oprab 7403  df-mpo 7404  df-er 8713  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-rp 13001  df-ico 13359  df-psmet 21292

This theorem is referenced by:  metustfbas  24481  metust  24482