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

Step	Hyp	Ref	Expression
1		eqid 2454	. . . . 5 |- ( Base ` W ) = ( Base ` W )
2		eqid 2454	. . . . 5 |- (UnifSt ` W ) = (UnifSt ` W )
3		uspreg.1	. . . . 5 |- J = ( TopOpen ` W )
4	1, 2, 3	isusp 20930	. . . 4 |- ( W e. UnifSp <-> ( (UnifSt ` W ) e. (UnifOn ` ( Base ` W ) ) /\ J = (unifTop ` (UnifSt ` W ) ) ) )
5	4	simprbi 462	. . 3 |- ( W e. UnifSp -> J = (unifTop ` (UnifSt ` W ) ) )
6	5	adantr 463	. 2 |- ( ( W e. UnifSp /\ J e. Haus ) -> J = (unifTop ` (UnifSt ` W ) ) )
7	4	simplbi 458	. . . 4 |- ( W e. UnifSp -> (UnifSt ` W ) e. (UnifOn ` ( Base ` W ) ) )
8	7	adantr 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 )
10	6, 9	eqeltrrd 2543	. . 3 |- ( ( W e. UnifSp /\ J e. Haus ) -> (unifTop ` (UnifSt ` W ) ) e. Haus )
11		eqid 2454	. . . 4 |- (unifTop ` (UnifSt ` W ) ) = (unifTop ` (UnifSt ` W ) )
12	11	utopreg 20921	. . 3 |- ( ( (UnifSt ` W ) e. (UnifOn ` ( Base ` W ) ) /\ (unifTop ` (UnifSt ` W ) ) e. Haus ) -> (unifTop ` (UnifSt ` W ) ) e. Reg )
13	8, 10, 12	syl2anc 659	. 2 |- ( ( W e. UnifSp /\ J e. Haus ) -> (unifTop ` (UnifSt ` W ) ) e. Reg )
14	6, 13	eqeltrd 2542	1 |- ( ( W e. UnifSp /\ J e. Haus ) -> J e. Reg )

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