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Theorem topbas 21893
Description: A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
topbas (𝐽 ∈ Top → 𝐽 ∈ TopBases)

Proof of Theorem topbas
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inopn 21820 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑦𝐽) → (𝑥𝑦) ∈ 𝐽)
213expb 1122 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) → (𝑥𝑦) ∈ 𝐽)
3 simpr 488 . . . . . . 7 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → 𝑧 ∈ (𝑥𝑦))
4 ssid 3937 . . . . . . 7 (𝑥𝑦) ⊆ (𝑥𝑦)
53, 4jctir 524 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
6 eleq2 2827 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑧𝑤𝑧 ∈ (𝑥𝑦)))
7 sseq1 3940 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑤 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
86, 7anbi12d 634 . . . . . . 7 (𝑤 = (𝑥𝑦) → ((𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))))
98rspcev 3549 . . . . . 6 (((𝑥𝑦) ∈ 𝐽 ∧ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
102, 5, 9syl2an2r 685 . . . . 5 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1110exp31 423 . . . 4 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → (𝑧 ∈ (𝑥𝑦) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))
1211ralrimdv 3110 . . 3 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1312ralrimivv 3112 . 2 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
14 isbasis2g 21869 . 2 (𝐽 ∈ Top → (𝐽 ∈ TopBases ↔ ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1513, 14mpbird 260 1 (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  wral 3062  wrex 3063  cin 3879  wss 3880  Topctop 21814  TopBasesctb 21866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5206
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-tru 1546  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-in 3887  df-ss 3897  df-pw 4529  df-uni 4834  df-top 21815  df-bases 21867
This theorem is referenced by:  resttop  22081  dis1stc  22420  txtop  22490  onpsstopbas  34382
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