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Theorem topontopi 22417
Description: A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypothesis
Ref Expression
topontopi.1 𝐽 ∈ (TopOnβ€˜π΅)
Assertion
Ref Expression
topontopi 𝐽 ∈ Top

Proof of Theorem topontopi
StepHypRef Expression
1 topontopi.1 . 2 𝐽 ∈ (TopOnβ€˜π΅)
2 topontop 22415 . 2 (𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐽 ∈ Top)
31, 2ax-mp 5 1 𝐽 ∈ Top
Colors of variables: wff setvar class
Syntax hints:   ∈ wcel 2107  β€˜cfv 6544  Topctop 22395  TopOnctopon 22412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-topon 22413
This theorem is referenced by:  sn0top  22502  indistop  22505  letop  22710  dfac14  23122  cnfldtop  24300  sszcld  24333  iitop  24396  limccnp2  25409  cxpcn3  26256  lmlim  32927  pnfneige0  32931  sxbrsigalem4  33286  knoppcnlem10  35378  poimir  36521  islptre  44335  fourierdlem62  44884
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