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Mirrors > Home > ILE Home > Th. List > xmeterval | 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 12891 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | ffn 5331 | . . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
3 | elpreima 5598 | . . 3 ⊢ (𝐷 Fn (𝑋 × 𝑋) → (〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ))) | |
4 | 1, 2, 3 | 3syl 17 | . 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 3982 | . . 3 ⊢ (𝐴 ∼ 𝐵 ↔ 𝐴(◡𝐷 “ ℝ)𝐵) |
7 | df-br 3977 | . . 3 ⊢ (𝐴(◡𝐷 “ ℝ)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) | |
8 | 6, 7 | bitri 183 | . 2 ⊢ (𝐴 ∼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) |
9 | df-3an 969 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ)) | |
10 | opelxp 4628 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) | |
11 | 10 | bicomi 131 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
12 | df-ov 5839 | . . . . 5 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
13 | 12 | eleq1i 2230 | . . . 4 ⊢ ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ) |
14 | 11, 13 | anbi12i 456 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
15 | 9, 14 | bitri 183 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
16 | 4, 8, 15 | 3bitr4g 222 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 〈cop 3573 class class class wbr 3976 × cxp 4596 ◡ccnv 4597 “ cima 4601 Fn wfn 5177 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 ℝ*cxr 7923 ∞Metcxmet 12521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-xmet 12529 |
This theorem is referenced by: xmeter 12977 xmetec 12978 xmetresbl 12981 |
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