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Theorem xmeterval 23127
 Description: Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
Hypothesis
Ref Expression
xmeter.1 = (𝐷 “ ℝ)
Assertion
Ref Expression
xmeterval (𝐷 ∈ (∞Met‘𝑋) → (𝐴 𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ)))

Proof of Theorem xmeterval
StepHypRef Expression
1 xmetf 23024 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2 ffn 6499 . . 3 (𝐷:(𝑋 × 𝑋)⟶ℝ*𝐷 Fn (𝑋 × 𝑋))
3 elpreima 6820 . . 3 (𝐷 Fn (𝑋 × 𝑋) → (⟨𝐴, 𝐵⟩ ∈ (𝐷 “ ℝ) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝐴, 𝐵⟩) ∈ ℝ)))
41, 2, 33syl 18 . 2 (𝐷 ∈ (∞Met‘𝑋) → (⟨𝐴, 𝐵⟩ ∈ (𝐷 “ ℝ) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝐴, 𝐵⟩) ∈ ℝ)))
5 xmeter.1 . . . 4 = (𝐷 “ ℝ)
65breqi 5039 . . 3 (𝐴 𝐵𝐴(𝐷 “ ℝ)𝐵)
7 df-br 5034 . . 3 (𝐴(𝐷 “ ℝ)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐷 “ ℝ))
86, 7bitri 278 . 2 (𝐴 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐷 “ ℝ))
9 df-3an 1087 . . 3 ((𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ))
10 opelxp 5561 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) ↔ (𝐴𝑋𝐵𝑋))
1110bicomi 227 . . . 4 ((𝐴𝑋𝐵𝑋) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
12 df-ov 7154 . . . . 5 (𝐴𝐷𝐵) = (𝐷‘⟨𝐴, 𝐵⟩)
1312eleq1i 2843 . . . 4 ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐷‘⟨𝐴, 𝐵⟩) ∈ ℝ)
1411, 13anbi12i 630 . . 3 (((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝐴, 𝐵⟩) ∈ ℝ))
159, 14bitri 278 . 2 ((𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝐴, 𝐵⟩) ∈ ℝ))
164, 8, 153bitr4g 318 1 (𝐷 ∈ (∞Met‘𝑋) → (𝐴 𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ⟨cop 4529   class class class wbr 5033   × cxp 5523  ◡ccnv 5524   “ cima 5528   Fn wfn 6331  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  ℝcr 10567  ℝ*cxr 10705  ∞Metcxmet 20144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10624  ax-resscn 10625 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8419  df-xr 10710  df-xmet 20152 This theorem is referenced by:  xmeter  23128  xmetec  23129  xmetresbl  23132  xrsblre  23505  isbndx  35493
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