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Theorem topnvalg 11816
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 2644 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
2 restfn 11808 . . . 4  |-t  Fn  ( _V  X.  _V )
3 topnval.2 . . . . 5  |-  J  =  (TopSet `  W )
4 tsetslid 11793 . . . . . 6  |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
54slotex 11670 . . . . 5  |-  ( W  e.  V  ->  (TopSet `  W )  e.  _V )
63, 5syl5eqel 2181 . . . 4  |-  ( W  e.  V  ->  J  e.  _V )
7 topnval.1 . . . . 5  |-  B  =  ( Base `  W
)
8 baseslid 11699 . . . . . 6  |-  ( Base 
= Slot  ( Base `  ndx )  /\  ( Base `  ndx )  e.  NN )
98slotex 11670 . . . . 5  |-  ( W  e.  V  ->  ( Base `  W )  e. 
_V )
107, 9syl5eqel 2181 . . . 4  |-  ( W  e.  V  ->  B  e.  _V )
11 fnovex 5720 . . . 4  |-  ( (t  Fn  ( _V  X.  _V )  /\  J  e.  _V  /\  B  e.  _V )  ->  ( Jt  B )  e.  _V )
122, 6, 10, 11mp3an2i 1285 . . 3  |-  ( W  e.  V  ->  ( Jt  B )  e.  _V )
13 fveq2 5340 . . . . . 6  |-  ( w  =  W  ->  (TopSet `  w )  =  (TopSet `  W ) )
1413, 3syl6eqr 2145 . . . . 5  |-  ( w  =  W  ->  (TopSet `  w )  =  J )
15 fveq2 5340 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
1615, 7syl6eqr 2145 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  B )
1714, 16oveq12d 5708 . . . 4  |-  ( w  =  W  ->  (
(TopSet `  w )t  ( Base `  w ) )  =  ( Jt  B ) )
18 df-topn 11807 . . . 4  |-  TopOpen  =  ( w  e.  _V  |->  ( (TopSet `  w )t  ( Base `  w ) ) )
1917, 18fvmptg 5415 . . 3  |-  ( ( W  e.  _V  /\  ( Jt  B )  e.  _V )  ->  ( TopOpen `  W
)  =  ( Jt  B ) )
201, 12, 19syl2anc 404 . 2  |-  ( W  e.  V  ->  ( TopOpen
`  W )  =  ( Jt  B ) )
2120eqcomd 2100 1  |-  ( W  e.  V  ->  ( Jt  B )  =  (
TopOpen `  W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296    e. wcel 1445   _Vcvv 2633    X. cxp 4465    Fn wfn 5044   ` cfv 5049  (class class class)co 5690   Basecbs 11643  TopSetcts 11711   ↾t crest 11804   TopOpenctopn 11805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-cnex 7533  ax-resscn 7534  ax-1re 7536  ax-addrcl 7539
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-5 8582  df-6 8583  df-7 8584  df-8 8585  df-9 8586  df-ndx 11646  df-slot 11647  df-base 11649  df-tset 11724  df-rest 11806  df-topn 11807
This theorem is referenced by:  topnidg  11817  topnpropgd  11818
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