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Theorem distopon 22855
Description: The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
distopon (𝐴 ∈ 𝑉 β†’ 𝒫 𝐴 ∈ (TopOnβ€˜π΄))

Proof of Theorem distopon
StepHypRef Expression
1 distop 22853 . 2 (𝐴 ∈ 𝑉 β†’ 𝒫 𝐴 ∈ Top)
2 unipw 5443 . . 3 βˆͺ 𝒫 𝐴 = 𝐴
32eqcomi 2735 . 2 𝐴 = βˆͺ 𝒫 𝐴
4 istopon 22769 . 2 (𝒫 𝐴 ∈ (TopOnβ€˜π΄) ↔ (𝒫 𝐴 ∈ Top ∧ 𝐴 = βˆͺ 𝒫 𝐴))
51, 3, 4sylanblrc 589 1 (𝐴 ∈ 𝑉 β†’ 𝒫 𝐴 ∈ (TopOnβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  π’« cpw 4597  βˆͺ cuni 4902  β€˜cfv 6537  Topctop 22750  TopOnctopon 22767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-top 22751  df-topon 22768
This theorem is referenced by:  sn0topon  22856  toponmre  22952  cndis  23150  txdis1cn  23494  xkofvcn  23543  distgp  23958  efmndtmd  23960  symgtgp  23965  cnfdmsn  45167
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