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Theorem bastop 21600
 Description: Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
bastop (𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))

Proof of Theorem bastop
StepHypRef Expression
1 tgtop 21592 . 2 (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
2 tgcl 21588 . . 3 (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
3 eleq1 2877 . . 3 ((topGen‘𝐵) = 𝐵 → ((topGen‘𝐵) ∈ Top ↔ 𝐵 ∈ Top))
42, 3syl5ibcom 248 . 2 (𝐵 ∈ TopBases → ((topGen‘𝐵) = 𝐵𝐵 ∈ Top))
51, 4impbid2 229 1 (𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ‘cfv 6327  topGenctg 16710  Topctop 21512  TopBasesctb 21564 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6286  df-fun 6329  df-fv 6335  df-topgen 16716  df-top 21513  df-bases 21565 This theorem is referenced by: (None)
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