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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 14233 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | ffn 5379 | . . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
3 | elpreima 5650 | . . 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 4023 | . . 3 ⊢ (𝐴 ∼ 𝐵 ↔ 𝐴(◡𝐷 “ ℝ)𝐵) |
7 | df-br 4018 | . . 3 ⊢ (𝐴(◡𝐷 “ ℝ)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) | |
8 | 6, 7 | bitri 184 | . 2 ⊢ (𝐴 ∼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) |
9 | df-3an 981 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ)) | |
10 | opelxp 4670 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) | |
11 | 10 | bicomi 132 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
12 | df-ov 5893 | . . . . 5 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
13 | 12 | eleq1i 2254 | . . . 4 ⊢ ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ) |
14 | 11, 13 | anbi12i 460 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
15 | 9, 14 | bitri 184 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
16 | 4, 8, 15 | 3bitr4g 223 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 979 = wceq 1363 ∈ wcel 2159 〈cop 3609 class class class wbr 4017 × cxp 4638 ◡ccnv 4639 “ cima 4643 Fn wfn 5225 ⟶wf 5226 ‘cfv 5230 (class class class)co 5890 ℝcr 7827 ℝ*cxr 8008 ∞Metcxmet 13809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-ral 2472 df-rex 2473 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-fv 5238 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-map 6667 df-pnf 8011 df-mnf 8012 df-xr 8013 df-xmet 13817 |
This theorem is referenced by: xmeter 14319 xmetec 14320 xmetresbl 14323 |
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