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Theorem topontopi 21518
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 21516 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
31, 2ax-mp 5 1 𝐽 ∈ Top
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  cfv 6344  Topctop 21496  TopOnctopon 21513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-topon 21514
This theorem is referenced by:  sn0top  21602  indistop  21605  letop  21809  dfac14  22221  cnfldtop  23387  sszcld  23420  iitop  23483  limccnp2  24493  cxpcn3  25335  lmlim  31217  pnfneige0  31221  sxbrsigalem4  31572  knoppcnlem10  33868  poimir  35002  islptre  42127  fourierdlem62  42676
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