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Theorem topfneec 33587
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 33585 . . . 4 Er V
3 errel 8291 . . . 4 ( Er V → Rel )
42, 3ax-mp 5 . . 3 Rel
5 relelec 8327 . . 3 (Rel → (𝐴 ∈ [𝐽] 𝐽 𝐴))
64, 5ax-mp 5 . 2 (𝐴 ∈ [𝐽] 𝐽 𝐴)
74brrelex2i 5607 . . . 4 (𝐽 𝐴𝐴 ∈ V)
87a1i 11 . . 3 (𝐽 ∈ Top → (𝐽 𝐴𝐴 ∈ V))
9 eleq1 2904 . . . . . . 7 ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top))
109biimparc 480 . . . . . 6 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top)
11 tgclb 21494 . . . . . 6 (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top)
1210, 11sylibr 235 . . . . 5 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases)
13 elex 3517 . . . . 5 (𝐴 ∈ TopBases → 𝐴 ∈ V)
1412, 13syl 17 . . . 4 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V)
1514ex 413 . . 3 (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽𝐴 ∈ V))
161fneval 33584 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴)))
17 tgtop 21497 . . . . . . . 8 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
1817eqeq1d 2827 . . . . . . 7 (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴)))
19 eqcom 2832 . . . . . . 7 (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)
2018, 19syl6bb 288 . . . . . 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 280 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
2322ex 413 . . 3 (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽)))
248, 15, 23pm5.21ndd 381 . 2 (𝐽 ∈ Top → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
256, 24syl5bb 284 1 (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ↔ (topGen‘𝐴) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  Vcvv 3499  cin 3938   class class class wbr 5062  ccnv 5552  Rel wrel 5558  cfv 6351   Er wer 8279  [cec 8280  topGenctg 16703  Topctop 21417  TopBasesctb 21469  Fnecfne 33568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fv 6359  df-er 8282  df-ec 8284  df-topgen 16709  df-top 21418  df-bases 21470  df-fne 33569
This theorem is referenced by: (None)
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