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Theorem topnvalg 12142
Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
Hypotheses
Ref Expression
topnval.1  |-  B  =  ( Base `  W
)
topnval.2  |-  J  =  (TopSet `  W )
Assertion
Ref Expression
topnvalg  |-  ( W  e.  V  ->  ( Jt  B )  =  (
TopOpen `  W ) )

Proof of Theorem topnvalg
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 2697 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
2 restfn 12134 . . . 4  |-t  Fn  ( _V  X.  _V )
3 topnval.2 . . . . 5  |-  J  =  (TopSet `  W )
4 tsetslid 12119 . . . . . 6  |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
54slotex 11996 . . . . 5  |-  ( W  e.  V  ->  (TopSet `  W )  e.  _V )
63, 5eqeltrid 2226 . . . 4  |-  ( W  e.  V  ->  J  e.  _V )
7 topnval.1 . . . . 5  |-  B  =  ( Base `  W
)
8 baseslid 12025 . . . . . 6  |-  ( Base 
= Slot  ( Base `  ndx )  /\  ( Base `  ndx )  e.  NN )
98slotex 11996 . . . . 5  |-  ( W  e.  V  ->  ( Base `  W )  e. 
_V )
107, 9eqeltrid 2226 . . . 4  |-  ( W  e.  V  ->  B  e.  _V )
11 fnovex 5804 . . . 4  |-  ( (t  Fn  ( _V  X.  _V )  /\  J  e.  _V  /\  B  e.  _V )  ->  ( Jt  B )  e.  _V )
122, 6, 10, 11mp3an2i 1320 . . 3  |-  ( W  e.  V  ->  ( Jt  B )  e.  _V )
13 fveq2 5421 . . . . . 6  |-  ( w  =  W  ->  (TopSet `  w )  =  (TopSet `  W ) )
1413, 3syl6eqr 2190 . . . . 5  |-  ( w  =  W  ->  (TopSet `  w )  =  J )
15 fveq2 5421 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
1615, 7syl6eqr 2190 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  B )
1714, 16oveq12d 5792 . . . 4  |-  ( w  =  W  ->  (
(TopSet `  w )t  ( Base `  w ) )  =  ( Jt  B ) )
18 df-topn 12133 . . . 4  |-  TopOpen  =  ( w  e.  _V  |->  ( (TopSet `  w )t  ( Base `  w ) ) )
1917, 18fvmptg 5497 . . 3  |-  ( ( W  e.  _V  /\  ( Jt  B )  e.  _V )  ->  ( TopOpen `  W
)  =  ( Jt  B ) )
201, 12, 19syl2anc 408 . 2  |-  ( W  e.  V  ->  ( TopOpen
`  W )  =  ( Jt  B ) )
2120eqcomd 2145 1  |-  ( W  e.  V  ->  ( Jt  B )  =  (
TopOpen `  W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   _Vcvv 2686    X. cxp 4537    Fn wfn 5118   ` cfv 5123  (class class class)co 5774   Basecbs 11969  TopSetcts 12037   ↾t crest 12130   TopOpenctopn 12131
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 11972  df-slot 11973  df-base 11975  df-tset 12050  df-rest 12132  df-topn 12133
This theorem is referenced by:  topnidg  12143  topnpropgd  12144
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