Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  topfneec Structured version   Visualization version   GIF version

Theorem topfneec 32862
Description: A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
topfneec.1 = (Fne ∩ Fne)
Assertion
Ref Expression
topfneec (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ↔ (topGen‘𝐴) = 𝐽))

Proof of Theorem topfneec
StepHypRef Expression
1 topfneec.1 . . . . 5 = (Fne ∩ Fne)
21fneer 32860 . . . 4 Er V
3 errel 7991 . . . 4 ( Er V → Rel )
42, 3ax-mp 5 . . 3 Rel
5 relelec 8025 . . 3 (Rel → (𝐴 ∈ [𝐽] 𝐽 𝐴))
64, 5ax-mp 5 . 2 (𝐴 ∈ [𝐽] 𝐽 𝐴)
74brrelex2i 5364 . . . 4 (𝐽 𝐴𝐴 ∈ V)
87a1i 11 . . 3 (𝐽 ∈ Top → (𝐽 𝐴𝐴 ∈ V))
9 eleq1 2866 . . . . . . 7 ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top))
109biimparc 472 . . . . . 6 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top)
11 tgclb 21103 . . . . . 6 (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top)
1210, 11sylibr 226 . . . . 5 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases)
13 elex 3400 . . . . 5 (𝐴 ∈ TopBases → 𝐴 ∈ V)
1412, 13syl 17 . . . 4 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V)
1514ex 402 . . 3 (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽𝐴 ∈ V))
161fneval 32859 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴)))
17 tgtop 21106 . . . . . . . 8 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
1817eqeq1d 2801 . . . . . . 7 (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴)))
19 eqcom 2806 . . . . . . 7 (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)
2018, 19syl6bb 279 . . . . . 6 (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
2120adantr 473 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
2216, 21bitrd 271 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
2322ex 402 . . 3 (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽)))
248, 15, 23pm5.21ndd 371 . 2 (𝐽 ∈ Top → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
256, 24syl5bb 275 1 (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ↔ (topGen‘𝐴) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  cin 3768   class class class wbr 4843  ccnv 5311  Rel wrel 5317  cfv 6101   Er wer 7979  [cec 7980  topGenctg 16413  Topctop 21026  TopBasesctb 21078  Fnecfne 32843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109  df-er 7982  df-ec 7984  df-topgen 16419  df-top 21027  df-bases 21079  df-fne 32844
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator