Theorem metustss 24526

Description: Range of the elements of the filter base generated by the metric 𝐷. (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
metustss	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))

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

Proof of Theorem metustss

Step	Hyp	Ref	Expression
1		metust.1	. . . 4 ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
2		cnvimass 6041	. . . . . . . . 9 ⊢ (◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
3		psmetf 24281	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
4	2, 3	fssdm 6681	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
5	4	ad2antrr 727	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
6		cnvexg 7868	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
7		imaexg 7857	. . . . . . . . 9 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V)
8		elpwg 4545	. . . . . . . . 9 ⊢ ((◡𝐷 “ (0[,)𝑎)) ∈ V → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
9	6, 7, 8	3syl 18	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
10	9	ad2antrr 727	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
11	5, 10	mpbird 257	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
12	11	ralrimiva 3130	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ∀𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
13		eqid 2737	. . . . . 6 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
14	13	rnmptss 7069	. . . . 5 ⊢ (∀𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) → ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋))
15	12, 14	syl 17	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋))
16	1, 15	eqsstrid 3961	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
17		simpr 484	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝐹)
18	16, 17	sseldd 3923	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝒫 (𝑋 × 𝑋))
19	18	elpwid 4551	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890  𝒫 cpw 4542   ↦ cmpt 5167   × cxp 5622  ^◡ccnv 5623  ran crn 5625   “ cima 5627  ‘cfv 6492  (class class class)co 7360  0cc0 11029  ℝ^*cxr 11169  ℝ⁺crp 12933  [,)cico 13291  PsMetcpsmet 21328

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8768  df-xr 11174  df-psmet 21336

This theorem is referenced by:  metustrel  24527  metustsym  24530