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Theorem ustelimasn 24082
Description: Any point 𝐴 is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
Assertion
Ref Expression
ustelimasn ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ (𝑉 β€œ {𝐴}))
Proof of Theorem ustelimasn
StepHypRef Expression
1 simp3 1135 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
2 ustdiag 24068 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ( I β†Ύ 𝑋) βŠ† 𝑉)
323adant3 1129 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ ( I β†Ύ 𝑋) βŠ† 𝑉)
4 opelidres 5987 . . . . 5 (𝐴 ∈ 𝑋 β†’ (⟨𝐴, 𝐴⟩ ∈ ( I β†Ύ 𝑋) ↔ 𝐴 ∈ 𝑋))
54ibir 268 . . . 4 (𝐴 ∈ 𝑋 β†’ ⟨𝐴, 𝐴⟩ ∈ ( I β†Ύ 𝑋))
653ad2ant3 1132 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ ⟨𝐴, 𝐴⟩ ∈ ( I β†Ύ 𝑋))
73, 6sseldd 3978 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ ⟨𝐴, 𝐴⟩ ∈ 𝑉)
8 elimasng 6081 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ (𝑉 β€œ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
98anidms 566 . . 3 (𝐴 ∈ 𝑋 β†’ (𝐴 ∈ (𝑉 β€œ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
109biimpar 477 . 2 ((𝐴 ∈ 𝑋 ∧ ⟨𝐴, 𝐴⟩ ∈ 𝑉) β†’ 𝐴 ∈ (𝑉 β€œ {𝐴}))
111, 7, 10syl2anc 583 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ (𝑉 β€œ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1084   ∈ wcel 2098   βŠ† wss 3943  {csn 4623  βŸ¨cop 4629   I cid 5566   β†Ύ cres 5671   β€œ cima 5672  β€˜cfv 6537  UnifOncust 24059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fv 6545  df-ust 24060
This theorem is referenced by: (None)
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