Theorem topnvalg 12135

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 -> ( J ↾t B ) = ( TopOpen ` W ) )

Proof of Theorem topnvalg

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2697	. . 3 |- ( W e. V -> W e. _V )
2		restfn 12127	. . . 4 |- ↾t Fn ( _V X. _V )
3		topnval.2	. . . . 5 |- J = (TopSet ` W )
4		tsetslid 12112	. . . . . 6 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
5	4	slotex 11989	. . . . 5 |- ( W e. V -> (TopSet ` W ) e. _V )
6	3, 5	eqeltrid 2226	. . . 4 |- ( W e. V -> J e. _V )
7		topnval.1	. . . . 5 |- B = ( Base ` W )
8		baseslid 12018	. . . . . 6 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
9	8	slotex 11989	. . . . 5 |- ( W e. V -> ( Base ` W ) e. _V )
10	7, 9	eqeltrid 2226	. . . 4 |- ( W e. V -> B e. _V )
11		fnovex 5804	. . . 4 |- ( ( ↾t Fn ( _V X. _V ) /\ J e. _V /\ B e. _V ) -> ( J ↾t B ) e. _V )
12	2, 6, 10, 11	mp3an2i 1320	. . 3 |- ( W e. V -> ( J ↾t B ) e. _V )
13		fveq2 5421	. . . . . 6 |- ( w = W -> (TopSet ` w ) = (TopSet ` W ) )
14	13, 3	syl6eqr 2190	. . . . 5 |- ( w = W -> (TopSet ` w ) = J )
15		fveq2 5421	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
16	15, 7	syl6eqr 2190	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
17	14, 16	oveq12d 5792	. . . 4 |- ( w = W -> ( (TopSet ` w ) ↾t ( Base ` w ) ) = ( J ↾t B ) )
18		df-topn 12126	. . . 4 |- TopOpen = ( w e. _V |-> ( (TopSet ` w ) ↾t ( Base ` w ) ) )
19	17, 18	fvmptg 5497	. . 3 |- ( ( W e. _V /\ ( J ↾t B ) e. _V ) -> ( TopOpen ` W ) = ( J ↾t B ) )
20	1, 12, 19	syl2anc 408	. 2 |- ( W e. V -> ( TopOpen ` W ) = ( J ↾t B ) )
21	20	eqcomd 2145	1 |- ( W e. V -> ( J ↾t B ) = ( TopOpen ` W ) )

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 11962  TopSetcts 12030   ↾_t crest 12123   TopOpenctopn 12124

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 7714  ax-resscn 7715  ax-1re 7717  ax-addrcl 7720

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 8724  df-2 8782  df-3 8783  df-4 8784  df-5 8785  df-6 8786  df-7 8787  df-8 8788  df-9 8789  df-ndx 11965  df-slot 11966  df-base 11968  df-tset 12043  df-rest 12125  df-topn 12126

This theorem is referenced by:  topnidg  12136  topnpropgd  12137