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Theorem topontopi 22287
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 22285 . 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 6500  Topctop 22265  TopOnctopon 22282
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-topon 22283
This theorem is referenced by:  sn0top  22372  indistop  22375  letop  22580  dfac14  22992  cnfldtop  24170  sszcld  24203  iitop  24266  limccnp2  25279  cxpcn3  26124  lmlim  32592  pnfneige0  32596  sxbrsigalem4  32951  knoppcnlem10  35018  poimir  36161  islptre  43950  fourierdlem62  44499
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