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Theorem ustref 20905
Description: Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
ustref  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A V A )

Proof of Theorem ustref
StepHypRef Expression
1 eqid 2402 . . . . 5  |-  A  =  A
2 resieq 5225 . . . . 5  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A (  _I  |`  X ) A  <->  A  =  A ) )
31, 2mpbiri 233 . . . 4  |-  ( ( A  e.  X  /\  A  e.  X )  ->  A (  _I  |`  X ) A )
43anidms 643 . . 3  |-  ( A  e.  X  ->  A
(  _I  |`  X ) A )
543ad2ant3 1020 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A
(  _I  |`  X ) A )
6 ustdiag 20895 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X )  C_  V )
76ssbrd 4435 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( A (  _I  |`  X ) A  ->  A V A ) )
873adant3 1017 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  ( A (  _I  |`  X ) A  ->  A V A ) )
95, 8mpd 15 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  A  e.  X )  ->  A V A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394    _I cid 4732    |` cres 4944   ` cfv 5525  UnifOncust 20886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-res 4954  df-iota 5489  df-fun 5527  df-fv 5533  df-ust 20887
This theorem is referenced by:  cstucnd  20971
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