MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustelimasn Structured version   Visualization version   GIF version

Theorem ustelimasn 22246
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 1132 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴𝑋)
2 ustdiag 22232 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
323adant3 1126 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ( I ↾ 𝑋) ⊆ 𝑉)
4 opelresi 5548 . . . . 5 (𝐴𝑋 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋) ↔ 𝐴𝑋))
54ibir 257 . . . 4 (𝐴𝑋 → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
653ad2ant3 1129 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
73, 6sseldd 3753 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → ⟨𝐴, 𝐴⟩ ∈ 𝑉)
8 elimasng 5631 . . . 4 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
98anidms 556 . . 3 (𝐴𝑋 → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
109biimpar 463 . 2 ((𝐴𝑋 ∧ ⟨𝐴, 𝐴⟩ ∈ 𝑉) → 𝐴 ∈ (𝑉 “ {𝐴}))
111, 7, 10syl2anc 573 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1071  wcel 2145  wss 3723  {csn 4317  cop 4323   I cid 5157  cres 5252  cima 5253  cfv 6030  UnifOncust 22223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-ust 22224
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator