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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 22134 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | ffn 6045 | . . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
3 | elpreima 6337 | . . 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 4659 | . . 3 ⊢ (𝐴 ∼ 𝐵 ↔ 𝐴(◡𝐷 “ ℝ)𝐵) |
7 | df-br 4654 | . . 3 ⊢ (𝐴(◡𝐷 “ ℝ)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) | |
8 | 6, 7 | bitri 264 | . 2 ⊢ (𝐴 ∼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) |
9 | df-3an 1039 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ)) | |
10 | opelxp 5146 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) | |
11 | 10 | bicomi 214 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
12 | df-ov 6653 | . . . . 5 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
13 | 12 | eleq1i 2692 | . . . 4 ⊢ ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ) |
14 | 11, 13 | anbi12i 733 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
15 | 9, 14 | bitri 264 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
16 | 4, 8, 15 | 3bitr4g 303 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 × cxp 5112 ◡ccnv 5113 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 ℝ*cxr 10073 ∞Metcxmt 19731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-xr 10078 df-xmet 19739 |
This theorem is referenced by: xmeter 22238 xmetec 22239 xmetresbl 22242 xrsblre 22614 isbndx 33581 |
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