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| 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 6100 | . . . . . . . . 9 ⊢ (◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷 | |
| 3 | psmetf 24316 | . . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
| 4 | 2, 3 | fssdm 6755 | . . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)) |
| 5 | 4 | ad2antrr 726 | . . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)) |
| 6 | cnvexg 7946 | . . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V) | |
| 7 | imaexg 7935 | . . . . . . . . 9 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V) | |
| 8 | elpwg 4603 | . . . . . . . . 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 726 | . . . . . . 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 3146 | . . . . 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 7143 | . . . . 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 4022 | . . 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 3984 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝒫 (𝑋 × 𝑋)) |
| 19 | 18 | elpwid 4609 | 1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 ran crn 5686 “ cima 5688 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℝ*cxr 11294 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 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-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-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-xr 11299 df-psmet 21356 |
| This theorem is referenced by: metustrel 24565 metustsym 24568 |
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