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Theorem ustelimasn 19913
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 990 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A  e.  X )
2 ustdiag 19899 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X )  C_  V )
323adant3 1008 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  (  _I  |`  X )  C_  V )
4 opelresi 5220 . . . . 5  |-  ( A  e.  X  ->  ( <. A ,  A >.  e.  (  _I  |`  X )  <-> 
A  e.  X ) )
54ibir 242 . . . 4  |-  ( A  e.  X  ->  <. A ,  A >.  e.  (  _I  |`  X ) )
653ad2ant3 1011 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  <. A ,  A >.  e.  (  _I  |`  X ) )
73, 6sseldd 3455 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  <. A ,  A >.  e.  V )
8 elimasng 5293 . . . 4  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A  e.  ( V " { A } )  <->  <. A ,  A >.  e.  V ) )
98anidms 645 . . 3  |-  ( A  e.  X  ->  ( A  e.  ( V " { A } )  <->  <. A ,  A >.  e.  V ) )
109biimpar 485 . 2  |-  ( ( A  e.  X  /\  <. A ,  A >.  e.  V )  ->  A  e.  ( V " { A } ) )
111, 7, 10syl2anc 661 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 965    e. wcel 1758    C_ wss 3426   {csn 3975   <.cop 3981    _I cid 4729    |` cres 4940   "cima 4941   ` cfv 5516  UnifOncust 19890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fv 5524  df-ust 19891
This theorem is referenced by: (None)
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