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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 21944 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | ffn 5958 | . . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
3 | elpreima 6245 | . . 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 4589 | . . 3 ⊢ (𝐴 ∼ 𝐵 ↔ 𝐴(◡𝐷 “ ℝ)𝐵) |
7 | df-br 4584 | . . 3 ⊢ (𝐴(◡𝐷 “ ℝ)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) | |
8 | 6, 7 | bitri 263 | . 2 ⊢ (𝐴 ∼ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝐷 “ ℝ)) |
9 | df-3an 1033 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ)) | |
10 | opelxp 5070 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) | |
11 | 10 | bicomi 213 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
12 | df-ov 6552 | . . . . 5 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
13 | 12 | eleq1i 2679 | . . . 4 ⊢ ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ) |
14 | 11, 13 | anbi12i 729 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
15 | 9, 14 | bitri 263 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝐴, 𝐵〉) ∈ ℝ)) |
16 | 4, 8, 15 | 3bitr4g 302 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 × cxp 5036 ◡ccnv 5037 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 ℝ*cxr 9952 ∞Metcxmt 19552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-xr 9957 df-xmet 19560 |
This theorem is referenced by: xmeter 22048 xmetec 22049 xmetresbl 22052 xrsblre 22422 isbndx 32751 |
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