Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > metustss | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
metust.1 | ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) |
Ref | Expression |
---|---|
metustss | ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metust.1 | . . . 4 ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) | |
2 | cnvimass 5404 | . . . . . . . . 9 ⊢ (◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷 | |
3 | psmetf 21921 | . . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
4 | fdm 5964 | . . . . . . . . . 10 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋)) | |
5 | 3, 4 | syl 17 | . . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
6 | 2, 5 | syl5sseq 3616 | . . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)) |
7 | 6 | ad2antrr 758 | . . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)) |
8 | cnvexg 7005 | . . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V) | |
9 | imaexg 6995 | . . . . . . . . 9 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V) | |
10 | elpwg 4116 | . . . . . . . . 9 ⊢ ((◡𝐷 “ (0[,)𝑎)) ∈ V → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))) | |
11 | 8, 9, 10 | 3syl 18 | . . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))) |
12 | 11 | ad2antrr 758 | . . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))) |
13 | 7, 12 | mpbird 246 | . . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋)) |
14 | 13 | ralrimiva 2949 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ∀𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋)) |
15 | eqid 2610 | . . . . . 6 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) | |
16 | 15 | rnmptss 6299 | . . . . 5 ⊢ (∀𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) → ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋)) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋)) |
18 | 1, 17 | syl5eqss 3612 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋)) |
19 | simpr 476 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝐹) | |
20 | 18, 19 | sseldd 3569 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝒫 (𝑋 × 𝑋)) |
21 | 20 | elpwid 4118 | 1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ↦ cmpt 4643 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℝ*cxr 9952 ℝ+crp 11708 [,)cico 12048 PsMetcpsmet 19551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-xr 9957 df-psmet 19559 |
This theorem is referenced by: metustrel 22167 metustsym 22170 |
Copyright terms: Public domain | W3C validator |