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Theorem topfneec 31520
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 31518 . . . 4 Er V
3 errel 7638 . . . 4 ( Er V → Rel )
42, 3ax-mp 5 . . 3 Rel
5 relelec 7674 . . 3 (Rel → (𝐴 ∈ [𝐽] 𝐽 𝐴))
64, 5ax-mp 5 . 2 (𝐴 ∈ [𝐽] 𝐽 𝐴)
74brrelex2i 5083 . . . 4 (𝐽 𝐴𝐴 ∈ V)
87a1i 11 . . 3 (𝐽 ∈ Top → (𝐽 𝐴𝐴 ∈ V))
9 eleq1 2676 . . . . . . 7 ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top))
109biimparc 503 . . . . . 6 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top)
11 tgclb 20585 . . . . . 6 (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top)
1210, 11sylibr 223 . . . . 5 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases)
13 elex 3185 . . . . 5 (𝐴 ∈ TopBases → 𝐴 ∈ V)
1412, 13syl 17 . . . 4 ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V)
1514ex 449 . . 3 (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽𝐴 ∈ V))
161fneval 31517 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴)))
17 tgtop 20588 . . . . . . . 8 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
1817eqeq1d 2612 . . . . . . 7 (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴)))
19 eqcom 2617 . . . . . . 7 (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)
2018, 19syl6bb 275 . . . . . 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 267 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
2322ex 449 . . 3 (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽)))
248, 15, 23pm5.21ndd 368 . 2 (𝐽 ∈ Top → (𝐽 𝐴 ↔ (topGen‘𝐴) = 𝐽))
256, 24syl5bb 271 1 (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ↔ (topGen‘𝐴) = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cin 3539   class class class wbr 4583  ccnv 5037  Rel wrel 5043  cfv 5804   Er wer 7626  [cec 7627  topGenctg 15921  Topctop 20517  TopBasesctb 20520  Fnecfne 31501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-er 7629  df-ec 7631  df-topgen 15927  df-top 20521  df-bases 20522  df-fne 31502
This theorem is referenced by: (None)
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