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Theorem toponcom 18651
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 18648 . . . 4  |-  ( K  e.  (TopOn `  U. J )  ->  U. J  =  U. K )
21eqcomd 2459 . . 3  |-  ( K  e.  (TopOn `  U. J )  ->  U. K  =  U. J )
32anim2i 569 . 2  |-  ( ( J  e.  Top  /\  K  e.  (TopOn `  U. J ) )  -> 
( J  e.  Top  /\ 
U. K  =  U. J ) )
4 istopon 18646 . 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 1370    e. wcel 1758   U.cuni 4189   ` cfv 5516   Topctop 18614  TopOnctopon 18615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-iota 5479  df-fun 5518  df-fv 5524  df-topon 18622
This theorem is referenced by:  kgencn3  19247
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