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Theorem ustelimasn 22350
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 1169 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴𝑋)
2 ustdiag 22336 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
323adant3 1163 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ( I ↾ 𝑋) ⊆ 𝑉)
4 opelidres 5617 . . . . 5 (𝐴𝑋 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋) ↔ 𝐴𝑋))
54ibir 260 . . . 4 (𝐴𝑋 → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
653ad2ant3 1166 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
73, 6sseldd 3797 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ⟨𝐴, 𝐴⟩ ∈ 𝑉)
8 elimasng 5706 . . . 4 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
98anidms 563 . . 3 (𝐴𝑋 → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
109biimpar 470 . 2 ((𝐴𝑋 ∧ ⟨𝐴, 𝐴⟩ ∈ 𝑉) → 𝐴 ∈ (𝑉 “ {𝐴}))
111, 7, 10syl2anc 580 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1108  wcel 2157  wss 3767  {csn 4366  cop 4372   I cid 5217  cres 5312  cima 5313  cfv 6099  UnifOncust 22327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fv 6107  df-ust 22328
This theorem is referenced by: (None)
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