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Theorem ustelimasn 20894
Description: Any point  A is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
Assertion
Ref Expression
ustelimasn  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A  e.  ( V " { A } ) )

Proof of Theorem ustelimasn
StepHypRef Expression
1 simp3 996 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A  e.  X )
2 ustdiag 20880 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X )  C_  V )
323adant3 1014 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  (  _I  |`  X )  C_  V )
4 opelresi 5273 . . . . 5  |-  ( A  e.  X  ->  ( <. A ,  A >.  e.  (  _I  |`  X )  <-> 
A  e.  X ) )
54ibir 242 . . . 4  |-  ( A  e.  X  ->  <. A ,  A >.  e.  (  _I  |`  X ) )
653ad2ant3 1017 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  <. A ,  A >.  e.  (  _I  |`  X ) )
73, 6sseldd 3490 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  <. A ,  A >.  e.  V )
8 elimasng 5351 . . . 4  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A  e.  ( V " { A } )  <->  <. A ,  A >.  e.  V ) )
98anidms 643 . . 3  |-  ( A  e.  X  ->  ( A  e.  ( V " { A } )  <->  <. A ,  A >.  e.  V ) )
109biimpar 483 . 2  |-  ( ( A  e.  X  /\  <. A ,  A >.  e.  V )  ->  A  e.  ( V " { A } ) )
111, 7, 10syl2anc 659 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A  e.  ( V " { A } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    e. wcel 1823    C_ wss 3461   {csn 4016   <.cop 4022    _I cid 4779    |` cres 4990   "cima 4991   ` cfv 5570  UnifOncust 20871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-ust 20872
This theorem is referenced by: (None)
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