ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  toponcomb Unicode version

Theorem toponcomb 13965
Description: Biconditional form of toponcom 13964. (Contributed by BJ, 5-Dec-2021.)
Assertion
Ref Expression
toponcomb  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  e.  (TopOn `  U. K )  <->  K  e.  (TopOn `  U. J ) ) )

Proof of Theorem toponcomb
StepHypRef Expression
1 toponcom 13964 . . . 4  |-  ( ( K  e.  Top  /\  J  e.  (TopOn `  U. K ) )  ->  K  e.  (TopOn `  U. J ) )
21ex 115 . . 3  |-  ( K  e.  Top  ->  ( J  e.  (TopOn `  U. K )  ->  K  e.  (TopOn `  U. J ) ) )
32adantl 277 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  e.  (TopOn `  U. K )  ->  K  e.  (TopOn `  U. J ) ) )
4 toponcom 13964 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  (TopOn `  U. J ) )  ->  J  e.  (TopOn `  U. K ) )
54ex 115 . . 3  |-  ( J  e.  Top  ->  ( K  e.  (TopOn `  U. J )  ->  J  e.  (TopOn `  U. K ) ) )
65adantr 276 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( K  e.  (TopOn `  U. J )  ->  J  e.  (TopOn `  U. K ) ) )
73, 6impbid 129 1  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  e.  (TopOn `  U. K )  <->  K  e.  (TopOn `  U. J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2160   U.cuni 3824   ` cfv 5232   Topctop 13934  TopOnctopon 13947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5234  df-fv 5240  df-topon 13948
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator