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Theorem bj-toptopon2 32065
 Description: A topology is the same thing as a topology on the union of its open sets. space. (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-toptopon2 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))

Proof of Theorem bj-toptopon2
StepHypRef Expression
1 eqid 2514 . 2 𝐽 = 𝐽
21toptopon 20451 1 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 194   ∈ wcel 1938  ∪ cuni 4270  ‘cfv 5689  Topctop 20420  TopOnctopon 20421 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-iota 5653  df-fun 5691  df-fv 5697  df-topon 20426 This theorem is referenced by:  bj-topontopon  32066  bj-toprntopon  32075
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