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

Theorem topontopi 22922
Description: A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypothesis
Ref Expression
topontopi.1 𝐽 ∈ (TopOn‘𝐵)
Assertion
Ref Expression
topontopi 𝐽 ∈ Top

Proof of Theorem topontopi
StepHypRef Expression
1 topontopi.1 . 2 𝐽 ∈ (TopOn‘𝐵)
2 topontop 22920 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
31, 2ax-mp 5 1 𝐽 ∈ Top
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  cfv 6560  Topctop 22900  TopOnctopon 22917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-topon 22918
This theorem is referenced by:  sn0top  23007  indistop  23010  letop  23215  dfac14  23627  cnfldtop  24805  sszcld  24840  iitop  24907  limccnp2  25928  cxpcn3  26792  lmlim  33947  pnfneige0  33951  sxbrsigalem4  34290  poimir  37661  islptre  45639  fourierdlem62  46188
  Copyright terms: Public domain W3C validator