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Theorem ustuqtop 20481
 Description: For a given uniform structure on a set , there is a unique topology such that the set is the filter of the neighborhoods of for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1
Assertion
Ref Expression
ustuqtop UnifOn TopOn
Distinct variable groups:   ,,   ,,,   ,,   ,,   ,
Allowed substitution hint:   ()

Proof of Theorem ustuqtop
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5864 . . . . . . 7
21eleq2d 2537 . . . . . 6
32cbvralv 3088 . . . . 5
4 eleq1 2539 . . . . . 6
54raleqbi1dv 3066 . . . . 5
63, 5syl5bbr 259 . . . 4
76cbvrabv 3112 . . 3
8 utopustuq.1 . . . 4
98ustuqtop0 20475 . . 3 UnifOn
108ustuqtop1 20476 . . 3 UnifOn
118ustuqtop2 20477 . . 3 UnifOn
128ustuqtop3 20478 . . 3 UnifOn
138ustuqtop4 20479 . . 3 UnifOn
148ustuqtop5 20480 . . 3 UnifOn
157, 9, 10, 11, 12, 13, 14neiptopreu 19397 . 2 UnifOn TopOn
169feqmptd 5918 . . . . 5 UnifOn
1716eqeq1d 2469 . . . 4 UnifOn
18 fvex 5874 . . . . . 6
1918rgenw 2825 . . . . 5
20 mpteqb 5962 . . . . 5
2119, 20ax-mp 5 . . . 4
2217, 21syl6bb 261 . . 3 UnifOn
2322reubidv 3046 . 2 UnifOn TopOn TopOn
2415, 23mpbid 210 1 UnifOn TopOn
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wceq 1379   wcel 1767  wral 2814  wreu 2816  crab 2818  cvv 3113  cpw 4010  csn 4027   cmpt 4505   crn 5000  cima 5002  cfv 5586  TopOnctopon 19159  cnei 19361  UnifOncust 20434 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-fin 7517  df-fi 7867  df-top 19163  df-topon 19166  df-ntr 19284  df-nei 19362  df-ust 20435 This theorem is referenced by: (None)
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