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Mirrors > Home > MPE Home > Th. List > metustel | Structured version Visualization version GIF version |
Description: Define a filter base 𝐹 generated by a metric 𝐷. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
metust.1 | ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) |
Ref | Expression |
---|---|
metustel | ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metust.1 | . . 3 ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) | |
2 | 1 | eleq2i 2831 | . 2 ⊢ (𝐵 ∈ 𝐹 ↔ 𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) |
3 | elex 3441 | . . . 4 ⊢ (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) → 𝐵 ∈ V) | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) → 𝐵 ∈ V)) |
5 | cnvexg 7723 | . . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V) | |
6 | imaexg 7714 | . . . . 5 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V) | |
7 | eleq1a 2835 | . . . . 5 ⊢ ((◡𝐷 “ (0[,)𝑎)) ∈ V → (𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V)) | |
8 | 5, 6, 7 | 3syl 18 | . . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V)) |
9 | 8 | rexlimdvw 3219 | . . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V)) |
10 | eqid 2739 | . . . . 5 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) | |
11 | 10 | elrnmpt 5842 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎)))) |
12 | 11 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ V → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎))))) |
13 | 4, 9, 12 | pm5.21ndd 384 | . 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎)))) |
14 | 2, 13 | syl5bb 286 | 1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∃wrex 3065 Vcvv 3423 ↦ cmpt 5151 ◡ccnv 5567 ran crn 5569 “ cima 5571 ‘cfv 6400 (class class class)co 7234 0cc0 10758 ℝ+crp 12615 [,)cico 12966 PsMetcpsmet 20379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-opab 5132 df-mpt 5152 df-xp 5574 df-rel 5575 df-cnv 5576 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 |
This theorem is referenced by: metustto 23482 metustid 23483 metustexhalf 23485 metustfbas 23486 cfilucfil 23488 metucn 23500 |
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