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Theorem toponcom 18494
Description: If  K is a topology on the base set of topology  J, then  J is a topology on the base of  K. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
toponcom  |-  ( ( J  e.  Top  /\  K  e.  (TopOn `  U. J ) )  ->  J  e.  (TopOn `  U. K ) )

Proof of Theorem toponcom
StepHypRef Expression
1 toponuni 18491 . . . 4  |-  ( K  e.  (TopOn `  U. J )  ->  U. J  =  U. K )
21eqcomd 2446 . . 3  |-  ( K  e.  (TopOn `  U. J )  ->  U. K  =  U. J )
32anim2i 566 . 2  |-  ( ( J  e.  Top  /\  K  e.  (TopOn `  U. J ) )  -> 
( J  e.  Top  /\ 
U. K  =  U. J ) )
4 istopon 18489 . 2  |-  ( J  e.  (TopOn `  U. K )  <->  ( J  e.  Top  /\  U. K  =  U. J ) )
53, 4sylibr 212 1  |-  ( ( J  e.  Top  /\  K  e.  (TopOn `  U. J ) )  ->  J  e.  (TopOn `  U. K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   U.cuni 4088   ` cfv 5415   Topctop 18457  TopOnctopon 18458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-topon 18465
This theorem is referenced by:  kgencn3  19090
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