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Theorem tgclb 21494
 Description: The property tgcl 21493 can be reversed: if the topology generated by 𝐵 is actually a topology, then 𝐵 must be a topological basis. This yields an alternative definition of TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
tgclb (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)

Proof of Theorem tgclb
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgcl 21493 . 2 (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
2 0opn 21428 . . . . . . . . . 10 ((topGen‘𝐵) ∈ Top → ∅ ∈ (topGen‘𝐵))
32elfvexd 6700 . . . . . . . . 9 ((topGen‘𝐵) ∈ Top → 𝐵 ∈ V)
4 bastg 21490 . . . . . . . . 9 (𝐵 ∈ V → 𝐵 ⊆ (topGen‘𝐵))
53, 4syl 17 . . . . . . . 8 ((topGen‘𝐵) ∈ Top → 𝐵 ⊆ (topGen‘𝐵))
65sselda 3970 . . . . . . 7 (((topGen‘𝐵) ∈ Top ∧ 𝑥𝐵) → 𝑥 ∈ (topGen‘𝐵))
75sselda 3970 . . . . . . 7 (((topGen‘𝐵) ∈ Top ∧ 𝑦𝐵) → 𝑦 ∈ (topGen‘𝐵))
86, 7anim12dan 618 . . . . . 6 (((topGen‘𝐵) ∈ Top ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)))
9 inopn 21423 . . . . . . 7 (((topGen‘𝐵) ∈ Top ∧ 𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → (𝑥𝑦) ∈ (topGen‘𝐵))
1093expb 1114 . . . . . 6 (((topGen‘𝐵) ∈ Top ∧ (𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵))) → (𝑥𝑦) ∈ (topGen‘𝐵))
118, 10syldan 591 . . . . 5 (((topGen‘𝐵) ∈ Top ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝑦) ∈ (topGen‘𝐵))
12 tg2 21489 . . . . . 6 (((𝑥𝑦) ∈ (topGen‘𝐵) ∧ 𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1312ralrimiva 3186 . . . . 5 ((𝑥𝑦) ∈ (topGen‘𝐵) → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1411, 13syl 17 . . . 4 (((topGen‘𝐵) ∈ Top ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1514ralrimivva 3195 . . 3 ((topGen‘𝐵) ∈ Top → ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
16 isbasis2g 21472 . . . 4 (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
173, 16syl 17 . . 3 ((topGen‘𝐵) ∈ Top → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1815, 17mpbird 258 . 2 ((topGen‘𝐵) ∈ Top → 𝐵 ∈ TopBases)
191, 18impbii 210 1 (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   ∈ wcel 2106  ∀wral 3142  ∃wrex 3143  Vcvv 3499   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  ‘cfv 6351  topGenctg 16703  Topctop 21417  TopBasesctb 21469 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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  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 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  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-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-iota 6311  df-fun 6353  df-fv 6359  df-topgen 16709  df-top 21418  df-bases 21470 This theorem is referenced by:  bastop2  21518  iocpnfordt  21739  icomnfordt  21740  iooordt  21741  tgcn  21776  tgcnp  21777  2ndcctbss  21979  2ndcomap  21982  dis2ndc  21984  flftg  22520  met2ndci  23047  xrtgioo  23329  topfneec  33588
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