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Theorem utoptopon 19944
Description: Topology induced by a uniform structure  U with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)
Assertion
Ref Expression
utoptopon  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  (TopOn `  X ) )

Proof of Theorem utoptopon
StepHypRef Expression
1 utoptop 19942 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  Top )
2 utopbas 19943 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  U. (unifTop `  U )
)
3 istopon 18663 . 2  |-  ( (unifTop `  U )  e.  (TopOn `  X )  <->  ( (unifTop `  U )  e.  Top  /\  X  =  U. (unifTop `  U ) ) )
41, 2, 3sylanbrc 664 1  |-  ( U  e.  (UnifOn `  X
)  ->  (unifTop `  U
)  e.  (TopOn `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   U.cuni 4200   ` cfv 5527   Topctop 18631  TopOnctopon 18632  UnifOncust 19907  unifTopcutop 19938
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-fv 5535  df-top 18636  df-topon 18639  df-ust 19908  df-utop 19939
This theorem is referenced by:  utop3cls  19959  tustps  19981
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