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Theorem ustelimasn 21836
 Description: Any point 𝐴 is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
Assertion
Ref Expression
ustelimasn ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))

Proof of Theorem ustelimasn
StepHypRef Expression
1 simp3 1056 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴𝑋)
2 ustdiag 21822 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
323adant3 1074 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ( I ↾ 𝑋) ⊆ 𝑉)
4 opelresi 5328 . . . . 5 (𝐴𝑋 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋) ↔ 𝐴𝑋))
54ibir 256 . . . 4 (𝐴𝑋 → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
653ad2ant3 1077 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
73, 6sseldd 3569 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ⟨𝐴, 𝐴⟩ ∈ 𝑉)
8 elimasng 5410 . . . 4 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
98anidms 675 . . 3 (𝐴𝑋 → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
109biimpar 501 . 2 ((𝐴𝑋 ∧ ⟨𝐴, 𝐴⟩ ∈ 𝑉) → 𝐴 ∈ (𝑉 “ {𝐴}))
111, 7, 10syl2anc 691 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   ∈ wcel 1977   ⊆ wss 3540  {csn 4125  ⟨cop 4131   I cid 4948   ↾ cres 5040   “ cima 5041  ‘cfv 5804  UnifOncust 21813 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 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-csb 3500  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-fv 5812  df-ust 21814 This theorem is referenced by: (None)
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