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Theorem ustelimasn 24145
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 24131 . . . 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 5991 . . . . 5 (𝐴 ∈ 𝑋 β†’ (⟨𝐴, 𝐴⟩ ∈ ( I β†Ύ 𝑋) ↔ 𝐴 ∈ 𝑋))
54ibir 267 . . . 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 3973 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑋) β†’ ⟨𝐴, 𝐴⟩ ∈ 𝑉)
8 elimasng 6087 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ (𝑉 β€œ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
98anidms 565 . . 3 (𝐴 ∈ 𝑋 β†’ (𝐴 ∈ (𝑉 β€œ {𝐴}) ↔ ⟨𝐴, 𝐴⟩ ∈ 𝑉))
109biimpar 476 . 2 ((𝐴 ∈ 𝑋 ∧ ⟨𝐴, 𝐴⟩ ∈ 𝑉) β†’ 𝐴 ∈ (𝑉 β€œ {𝐴}))
111, 7, 10syl2anc 582 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 3939  {csn 4624  βŸ¨cop 4630   I cid 5569   β†Ύ cres 5674   β€œ cima 5675  β€˜cfv 6543  UnifOncust 24122
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fv 6551  df-ust 24123
This theorem is referenced by: (None)
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