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Mirrors > Home > MPE Home > Th. List > xmeterval | Structured version Visualization version GIF version |
Description: Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
xmeter.1 | ⊢ ∼ = (◡𝐷 “ ℝ) |
Ref | Expression |
---|---|
xmeterval | ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetf 22936 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | ffn 6487 | . . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
3 | elpreima 6805 | . . 3 ⊢ (𝐷 Fn (𝑋 × 𝑋) → (〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ))) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ))) |
5 | xmeter.1 | . . . 4 ⊢ ∼ = (◡𝐷 “ ℝ) | |
6 | 5 | breqi 5036 | . . 3 ⊢ (𝐴 ∼ 𝐵 ↔ 𝐴(◡𝐷 “ ℝ)𝐵) |
7 | df-br 5031 | . . 3 ⊢ (𝐴(◡𝐷 “ ℝ)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) | |
8 | 6, 7 | bitri 278 | . 2 ⊢ (𝐴 ∼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) |
9 | df-3an 1086 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ)) | |
10 | opelxp 5555 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) | |
11 | 10 | bicomi 227 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
12 | df-ov 7138 | . . . . 5 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
13 | 12 | eleq1i 2880 | . . . 4 ⊢ ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ) |
14 | 11, 13 | anbi12i 629 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
15 | 9, 14 | bitri 278 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
16 | 4, 8, 15 | 3bitr4g 317 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 × cxp 5517 ◡ccnv 5518 “ cima 5522 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 ℝ*cxr 10663 ∞Metcxmet 20076 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-xr 10668 df-xmet 20084 |
This theorem is referenced by: xmeter 23040 xmetec 23041 xmetresbl 23044 xrsblre 23416 isbndx 35220 |
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