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Theorem topontopon 11969
 Description: A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
topontopon (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘ 𝐽))

Proof of Theorem topontopon
StepHypRef Expression
1 topontop 11963 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 toptopon2 11968 . 2 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
31, 2sylib 121 1 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘ 𝐽))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1448  ∪ cuni 3683  ‘cfv 5059  Topctop 11946  TopOnctopon 11959 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293 This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-topon 11960 This theorem is referenced by: (None)
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