MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  topbas Structured version   Visualization version   GIF version

Theorem topbas 21154
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 21081 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑦𝐽) → (𝑥𝑦) ∈ 𝐽)
213expb 1153 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) → (𝑥𝑦) ∈ 𝐽)
3 simpr 479 . . . . . . 7 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → 𝑧 ∈ (𝑥𝑦))
4 ssid 3848 . . . . . . 7 (𝑥𝑦) ⊆ (𝑥𝑦)
53, 4jctir 516 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
6 eleq2 2895 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑧𝑤𝑧 ∈ (𝑥𝑦)))
7 sseq1 3851 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑤 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
86, 7anbi12d 624 . . . . . . 7 (𝑤 = (𝑥𝑦) → ((𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))))
98rspcev 3526 . . . . . 6 (((𝑥𝑦) ∈ 𝐽 ∧ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
102, 5, 9syl2an2r 675 . . . . 5 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1110exp31 412 . . . 4 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → (𝑧 ∈ (𝑥𝑦) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))
1211ralrimdv 3177 . . 3 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1312ralrimivv 3179 . 2 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
14 isbasis2g 21130 . 2 (𝐽 ∈ Top → (𝐽 ∈ TopBases ↔ ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1513, 14mpbird 249 1 (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  wrex 3118  cin 3797  wss 3798  Topctop 21075  TopBasesctb 21127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-in 3805  df-ss 3812  df-pw 4382  df-uni 4661  df-top 21076  df-bases 21128
This theorem is referenced by:  resttop  21342  dis1stc  21680  txtop  21750  onpsstopbas  32957
  Copyright terms: Public domain W3C validator