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Theorem topontopon 12196
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  e.  (TopOn `  X
)  ->  J  e.  (TopOn `  U. J ) )

Proof of Theorem topontopon
StepHypRef Expression
1 topontop 12190 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 toptopon2 12195 . 2  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
31, 2sylib 121 1  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  (TopOn `  U. J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480   U.cuni 3736   ` cfv 5123   Topctop 12173  TopOnctopon 12186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-topon 12187
This theorem is referenced by: (None)
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