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Theorem xmeterval 22957
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 22854 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2 ffn 6510 . . 3 (𝐷:(𝑋 × 𝑋)⟶ℝ*𝐷 Fn (𝑋 × 𝑋))
3 elpreima 6823 . . 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 5068 . . 3 (𝐴 𝐵𝐴(𝐷 “ ℝ)𝐵)
7 df-br 5063 . . 3 (𝐴(𝐷 “ ℝ)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐷 “ ℝ))
86, 7bitri 276 . 2 (𝐴 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐷 “ ℝ))
9 df-3an 1083 . . 3 ((𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ ((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ))
10 opelxp 5589 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) ↔ (𝐴𝑋𝐵𝑋))
1110bicomi 225 . . . 4 ((𝐴𝑋𝐵𝑋) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
12 df-ov 7154 . . . . 5 (𝐴𝐷𝐵) = (𝐷‘⟨𝐴, 𝐵⟩)
1312eleq1i 2907 . . . 4 ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐷‘⟨𝐴, 𝐵⟩) ∈ ℝ)
1411, 13anbi12i 626 . . 3 (((𝐴𝑋𝐵𝑋) ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝐴, 𝐵⟩) ∈ ℝ))
159, 14bitri 276 . 2 ((𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝐴, 𝐵⟩) ∈ ℝ))
164, 8, 153bitr4g 315 1 (𝐷 ∈ (∞Met‘𝑋) → (𝐴 𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  cop 4569   class class class wbr 5062   × cxp 5551  ccnv 5552  cima 5556   Fn wfn 6346  wf 6347  cfv 6351  (class class class)co 7151  cr 10528  *cxr 10666  ∞Metcxmet 20446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8401  df-xr 10671  df-xmet 20454
This theorem is referenced by:  xmeter  22958  xmetec  22959  xmetresbl  22962  xrsblre  23334  isbndx  34928
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