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Theorem ustelimasn 22806
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 1135 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴𝑋)
2 ustdiag 22792 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
323adant3 1129 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ( I ↾ 𝑋) ⊆ 𝑉)
4 opelidres 5838 . . . . 5 (𝐴𝑋 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋) ↔ 𝐴𝑋))
54ibir 271 . . . 4 (𝐴𝑋 → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
653ad2ant3 1132 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
73, 6sseldd 3944 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ⟨𝐴, 𝐴⟩ ∈ 𝑉)
8 elimasng 5928 . . . 4 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
98anidms 570 . . 3 (𝐴𝑋 → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
109biimpar 481 . 2 ((𝐴𝑋 ∧ ⟨𝐴, 𝐴⟩ ∈ 𝑉) → 𝐴 ∈ (𝑉 “ {𝐴}))
111, 7, 10syl2anc 587 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084  wcel 2115  wss 3910  {csn 4540  cop 4546   I cid 5432  cres 5530  cima 5531  cfv 6328  UnifOncust 22783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fv 6336  df-ust 22784
This theorem is referenced by: (None)
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