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Theorem topfneec 31989
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 31987 . . . 4 Er V
3 errel 7696 . . . 4 ( Er V → Rel )
42, 3ax-mp 5 . . 3 Rel
5 relelec 7732 . . 3 (Rel → (𝐴 ∈ [𝐽] 𝐽 𝐴))
64, 5ax-mp 5 . 2 (𝐴 ∈ [𝐽] 𝐽 𝐴)
74brrelex2i 5119 . . . 4 (𝐽 𝐴𝐴 ∈ V)
87a1i 11 . . 3 (𝐽 ∈ Top → (𝐽 𝐴𝐴 ∈ V))
9 eleq1 2686 . . . . . . 7 ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top))
109biimparc 504 . . . . . 6 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top)
11 tgclb 20685 . . . . . 6 (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top)
1210, 11sylibr 224 . . . . 5 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases)
13 elex 3198 . . . . 5 (𝐴 ∈ TopBases → 𝐴 ∈ V)
1412, 13syl 17 . . . 4 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V)
1514ex 450 . . 3 (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽𝐴 ∈ V))
161fneval 31986 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴)))
17 tgtop 20688 . . . . . . . 8 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
1817eqeq1d 2623 . . . . . . 7 (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴)))
19 eqcom 2628 . . . . . . 7 (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)
2018, 19syl6bb 276 . . . . . 6 (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
2120adantr 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
2216, 21bitrd 268 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
2322ex 450 . . 3 (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽)))
248, 15, 23pm5.21ndd 369 . 2 (𝐽 ∈ Top → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
256, 24syl5bb 272 1 (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ↔ (topGen‘𝐴) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554   class class class wbr 4613  ccnv 5073  Rel wrel 5079  cfv 5847   Er wer 7684  [cec 7685  topGenctg 16019  Topctop 20617  TopBasesctb 20620  Fnecfne 31970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-er 7687  df-ec 7689  df-topgen 16025  df-top 20621  df-bases 20622  df-fne 31971
This theorem is referenced by: (None)
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