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

Theorem topnpropgd 12143
Description: The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
Hypotheses
Ref Expression
topnpropd.1  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
topnpropd.2  |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  L )
)
topnpropgd.k  |-  ( ph  ->  K  e.  V )
topnpropgd.l  |-  ( ph  ->  L  e.  W )
Assertion
Ref Expression
topnpropgd  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )

Proof of Theorem topnpropgd
StepHypRef Expression
1 topnpropd.2 . . 3  |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  L )
)
2 topnpropd.1 . . 3  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
31, 2oveq12d 5792 . 2  |-  ( ph  ->  ( (TopSet `  K
)t  ( Base `  K
) )  =  ( (TopSet `  L )t  ( Base `  L ) ) )
4 topnpropgd.k . . 3  |-  ( ph  ->  K  e.  V )
5 eqid 2139 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
6 eqid 2139 . . . 4  |-  (TopSet `  K )  =  (TopSet `  K )
75, 6topnvalg 12141 . . 3  |-  ( K  e.  V  ->  (
(TopSet `  K )t  ( Base `  K ) )  =  ( TopOpen `  K
) )
84, 7syl 14 . 2  |-  ( ph  ->  ( (TopSet `  K
)t  ( Base `  K
) )  =  (
TopOpen `  K ) )
9 topnpropgd.l . . 3  |-  ( ph  ->  L  e.  W )
10 eqid 2139 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
11 eqid 2139 . . . 4  |-  (TopSet `  L )  =  (TopSet `  L )
1210, 11topnvalg 12141 . . 3  |-  ( L  e.  W  ->  (
(TopSet `  L )t  ( Base `  L ) )  =  ( TopOpen `  L
) )
139, 12syl 14 . 2  |-  ( ph  ->  ( (TopSet `  L
)t  ( Base `  L
) )  =  (
TopOpen `  L ) )
143, 8, 133eqtr3d 2180 1  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   ` cfv 5123  (class class class)co 5774   Basecbs 11968  TopSetcts 12036   ↾t crest 12129   TopOpenctopn 12130
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-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-cnex 7718  ax-resscn 7719  ax-1re 7721  ax-addrcl 7724
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-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  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-int 3772  df-iun 3815  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-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-5 8789  df-6 8790  df-7 8791  df-8 8792  df-9 8793  df-ndx 11971  df-slot 11972  df-base 11974  df-tset 12049  df-rest 12131  df-topn 12132
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator