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Theorem topontopi 20942
 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 20940 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
31, 2ax-mp 5 1 𝐽 ∈ Top
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2139  ‘cfv 6049  Topctop 20920  TopOnctopon 20937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-topon 20938 This theorem is referenced by:  sn0top  21025  indistop  21028  letop  21232  dfac14  21643  cnfldtop  22808  sszcld  22841  iitop  22904  limccnp2  23875  cxpcn3  24709  lmlim  30323  pnfneige0  30327  sxbrsigalem4  30679  knoppcnlem10  32819  poimir  33773  islptre  40372  fourierdlem62  40906
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