Theorem ustelimasn 24380

Description: Any point 𝐴 is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)

Assertion

Ref	Expression
ustelimasn	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))

Proof of Theorem ustelimasn

Step	Hyp	Ref	Expression
1		simp3 1156	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
2		ustdiag 24366	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
3	2	3adant3 1150	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → ( I ↾ 𝑋) ⊆ 𝑉)
4		opelidres 5990	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ 𝑋))
5	4	ibir 271	. . . 4 ⊢ (𝐴 ∈ 𝑋 → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
6	5	3ad2ant3 1153	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → ⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝑋))
7	3, 6	sseldd 3938	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → ⟨𝐴, 𝐴⟩ ∈ 𝑉)
8		elimasng 6091	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
9	8	anidms 576	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ (𝑉 “ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
10	9	biimpar 482	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ ⟨𝐴, 𝐴⟩ ∈ 𝑉) → 𝐴 ∈ (𝑉 “ {𝐴}))
11	1, 7, 10	syl2anc 595	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1103   ∈ wcel 2143   ⊆ wss 3905  {csn 4589  ⟨cop 4595   I cid 5555   ↾ cres 5663   “ cima 5664  ‘cfv 6536  UnifOncust 24357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-ust 24358

This theorem is referenced by: (None)