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

Theorem tgtop 22996
Description: A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
tgtop (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)

Proof of Theorem tgtop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eltg3 22985 . . . 4 (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) ↔ ∃𝑦(𝑦𝐽𝑥 = 𝑦)))
2 simpr 484 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
3 uniopn 22919 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑦𝐽)
43adantr 480 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑦𝐽)
52, 4eqeltrd 2839 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑥 = 𝑦) → 𝑥𝐽)
65expl 457 . . . . 5 (𝐽 ∈ Top → ((𝑦𝐽𝑥 = 𝑦) → 𝑥𝐽))
76exlimdv 1931 . . . 4 (𝐽 ∈ Top → (∃𝑦(𝑦𝐽𝑥 = 𝑦) → 𝑥𝐽))
81, 7sylbid 240 . . 3 (𝐽 ∈ Top → (𝑥 ∈ (topGen‘𝐽) → 𝑥𝐽))
98ssrdv 4001 . 2 (𝐽 ∈ Top → (topGen‘𝐽) ⊆ 𝐽)
10 bastg 22989 . 2 (𝐽 ∈ Top → 𝐽 ⊆ (topGen‘𝐽))
119, 10eqssd 4013 1 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wss 3963   cuni 4912  cfv 6563  topGenctg 17484  Topctop 22915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-topgen 17490  df-top 22916
This theorem is referenced by:  eltop  22997  eltop2  22998  eltop3  22999  bastop  23004  tgtop11  23005  basgen  23011  tgfiss  23014  bastop1  23016  resttop  23184  dis1stc  23523  alexsubALTlem1  24071  xrtgioo  24842  topfne  36337  topfneec  36338  topfneec2  36339  dissneqlem  37323
  Copyright terms: Public domain W3C validator