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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 5949 | . . . . . . . . 9 ⊢ (◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷 | |
3 | psmetf 22916 | . . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
4 | 2, 3 | fssdm 6530 | . . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)) |
5 | 4 | ad2antrr 724 | . . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)) |
6 | cnvexg 7629 | . . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V) | |
7 | imaexg 7620 | . . . . . . . . 9 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V) | |
8 | elpwg 4542 | . . . . . . . . 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 724 | . . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))) |
11 | 5, 10 | mpbird 259 | . . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋)) |
12 | 11 | ralrimiva 3182 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ∀𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋)) |
13 | eqid 2821 | . . . . . 6 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) | |
14 | 13 | rnmptss 6886 | . . . . 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 4015 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋)) |
17 | simpr 487 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝐹) | |
18 | 16, 17 | sseldd 3968 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝒫 (𝑋 × 𝑋)) |
19 | 18 | elpwid 4550 | 1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 𝒫 cpw 4539 ↦ cmpt 5146 × cxp 5553 ◡ccnv 5554 ran crn 5556 “ cima 5558 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ℝ*cxr 10674 ℝ+crp 12390 [,)cico 12741 PsMetcpsmet 20529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-xr 10679 df-psmet 20537 |
This theorem is referenced by: metustrel 23162 metustsym 23165 |
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