Theorem topnvalg 12706

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 2750	. . 3 |- ( W e. V -> W e. _V )
2		restfn 12698	. . . 4 |- ↾t Fn ( _V X. _V )
3		topnval.2	. . . . 5 |- J = (TopSet ` W )
4		tsetslid 12649	. . . . . 6 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
5	4	slotex 12492	. . . . 5 |- ( W e. V -> (TopSet ` W ) e. _V )
6	3, 5	eqeltrid 2264	. . . 4 |- ( W e. V -> J e. _V )
7		topnval.1	. . . . 5 |- B = ( Base ` W )
8		baseslid 12522	. . . . . 6 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
9	8	slotex 12492	. . . . 5 |- ( W e. V -> ( Base ` W ) e. _V )
10	7, 9	eqeltrid 2264	. . . 4 |- ( W e. V -> B e. _V )
11		fnovex 5911	. . . 4 |- ( ( ↾t Fn ( _V X. _V ) /\ J e. _V /\ B e. _V ) -> ( J ↾t B ) e. _V )
12	2, 6, 10, 11	mp3an2i 1342	. . 3 |- ( W e. V -> ( J ↾t B ) e. _V )
13		fveq2 5517	. . . . . 6 |- ( w = W -> (TopSet ` w ) = (TopSet ` W ) )
14	13, 3	eqtr4di 2228	. . . . 5 |- ( w = W -> (TopSet ` w ) = J )
15		fveq2 5517	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
16	15, 7	eqtr4di 2228	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
17	14, 16	oveq12d 5896	. . . 4 |- ( w = W -> ( (TopSet ` w ) ↾t ( Base ` w ) ) = ( J ↾t B ) )
18		df-topn 12697	. . . 4 |- TopOpen = ( w e. _V |-> ( (TopSet ` w ) ↾t ( Base ` w ) ) )
19	17, 18	fvmptg 5595	. . 3 |- ( ( W e. _V /\ ( J ↾t B ) e. _V ) -> ( TopOpen ` W ) = ( J ↾t B ) )
20	1, 12, 19	syl2anc 411	. 2 |- ( W e. V -> ( TopOpen ` W ) = ( J ↾t B ) )
21	20	eqcomd 2183	1 |- ( W e. V -> ( J ↾t B ) = ( TopOpen ` W ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2739   X. cxp 4626   Fn wfn 5213   ` cfv 5218  (class class class)co 5878   Basecbs 12465  TopSetcts 12545   ↾_t crest 12694   TopOpenctopn 12695

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-5 8984  df-6 8985  df-7 8986  df-8 8987  df-9 8988  df-ndx 12468  df-slot 12469  df-base 12471  df-tset 12558  df-rest 12696  df-topn 12697

This theorem is referenced by:  topnidg  12707  topnpropgd  12708  mgptopng  13145