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Theorem topfneec 36356
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 36354 . . . 4 Er V
3 errel 8754 . . . 4 ( Er V → Rel )
42, 3ax-mp 5 . . 3 Rel
5 relelec 8792 . . 3 (Rel → (𝐴 ∈ [𝐽] 𝐽 𝐴))
64, 5ax-mp 5 . 2 (𝐴 ∈ [𝐽] 𝐽 𝐴)
74brrelex2i 5742 . . . 4 (𝐽 𝐴𝐴 ∈ V)
87a1i 11 . . 3 (𝐽 ∈ Top → (𝐽 𝐴𝐴 ∈ V))
9 eleq1 2829 . . . . . . 7 ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top))
109biimparc 479 . . . . . 6 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top)
11 tgclb 22977 . . . . . 6 (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top)
1210, 11sylibr 234 . . . . 5 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases)
13 elex 3501 . . . . 5 (𝐴 ∈ TopBases → 𝐴 ∈ V)
1412, 13syl 17 . . . 4 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V)
1514ex 412 . . 3 (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽𝐴 ∈ V))
161fneval 36353 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴)))
17 tgtop 22980 . . . . . . . 8 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
1817eqeq1d 2739 . . . . . . 7 (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴)))
19 eqcom 2744 . . . . . . 7 (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)
2018, 19bitrdi 287 . . . . . 6 (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
2120adantr 480 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽))
2216, 21bitrd 279 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
2322ex 412 . . 3 (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽)))
248, 15, 23pm5.21ndd 379 . 2 (𝐽 ∈ Top → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
256, 24bitrid 283 1 (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ↔ (topGen‘𝐴) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950   class class class wbr 5143  ccnv 5684  Rel wrel 5690  cfv 6561   Er wer 8742  [cec 8743  topGenctg 17482  Topctop 22899  TopBasesctb 22952  Fnecfne 36337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-er 8745  df-ec 8747  df-topgen 17488  df-top 22900  df-bases 22953  df-fne 36338
This theorem is referenced by: (None)
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