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Theorem uspreg 20943
Description: If a uniform space is Hausdorff, it is regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 4-Jan-2018.)
Hypothesis
Ref Expression
uspreg.1  |-  J  =  ( TopOpen `  W )
Assertion
Ref Expression
uspreg  |-  ( ( W  e. UnifSp  /\  J  e. 
Haus )  ->  J  e.  Reg )

Proof of Theorem uspreg
StepHypRef Expression
1 eqid 2454 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2454 . . . . 5  |-  (UnifSt `  W )  =  (UnifSt `  W )
3 uspreg.1 . . . . 5  |-  J  =  ( TopOpen `  W )
41, 2, 3isusp 20930 . . . 4  |-  ( W  e. UnifSp 
<->  ( (UnifSt `  W
)  e.  (UnifOn `  ( Base `  W )
)  /\  J  =  (unifTop `  (UnifSt `  W
) ) ) )
54simprbi 462 . . 3  |-  ( W  e. UnifSp  ->  J  =  (unifTop `  (UnifSt `  W )
) )
65adantr 463 . 2  |-  ( ( W  e. UnifSp  /\  J  e. 
Haus )  ->  J  =  (unifTop `  (UnifSt `  W
) ) )
74simplbi 458 . . . 4  |-  ( W  e. UnifSp  ->  (UnifSt `  W )  e.  (UnifOn `  ( Base `  W ) ) )
87adantr 463 . . 3  |-  ( ( W  e. UnifSp  /\  J  e. 
Haus )  ->  (UnifSt `  W )  e.  (UnifOn `  ( Base `  W
) ) )
9 simpr 459 . . . 4  |-  ( ( W  e. UnifSp  /\  J  e. 
Haus )  ->  J  e.  Haus )
106, 9eqeltrrd 2543 . . 3  |-  ( ( W  e. UnifSp  /\  J  e. 
Haus )  ->  (unifTop `  (UnifSt `  W )
)  e.  Haus )
11 eqid 2454 . . . 4  |-  (unifTop `  (UnifSt `  W ) )  =  (unifTop `  (UnifSt `  W
) )
1211utopreg 20921 . . 3  |-  ( ( (UnifSt `  W )  e.  (UnifOn `  ( Base `  W ) )  /\  (unifTop `  (UnifSt `  W
) )  e.  Haus )  ->  (unifTop `  (UnifSt `  W
) )  e.  Reg )
138, 10, 12syl2anc 659 . 2  |-  ( ( W  e. UnifSp  /\  J  e. 
Haus )  ->  (unifTop `  (UnifSt `  W )
)  e.  Reg )
146, 13eqeltrd 2542 1  |-  ( ( W  e. UnifSp  /\  J  e. 
Haus )  ->  J  e.  Reg )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ` cfv 5570   Basecbs 14716   TopOpenctopn 14911   Hauscha 19976   Regcreg 19977  UnifOncust 20868  unifTopcutop 20899  UnifStcuss 20922  UnifSpcusp 20923
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-rep 4550  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-3or 972  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-reu 2811  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-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  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-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-fin 7513  df-fi 7863  df-topgen 14933  df-top 19566  df-bases 19568  df-topon 19569  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-cn 19895  df-cnp 19896  df-reg 19984  df-tx 20229  df-ust 20869  df-utop 20900  df-usp 20926
This theorem is referenced by:  cnextucn  20972  rrhre  28233
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