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 -> ( J ↾t B ) = ( TopOpen ` W ) )

Proof of Theorem topnvalg

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	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 )
5	4	slotex 11670	. . . . 5 |- ( W e. V -> (TopSet ` W ) e. _V )
6	3, 5	syl5eqel 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 )
9	8	slotex 11670	. . . . 5 |- ( W e. V -> ( Base ` W ) e. _V )
10	7, 9	syl5eqel 2181	. . . 4 |- ( W e. V -> B e. _V )
11		fnovex 5720	. . . 4 |- ( ( ↾t Fn ( _V X. _V ) /\ J e. _V /\ B e. _V ) -> ( J ↾t B ) e. _V )
12	2, 6, 10, 11	mp3an2i 1285	. . 3 |- ( W e. V -> ( J ↾t B ) e. _V )
13		fveq2 5340	. . . . . 6 |- ( w = W -> (TopSet ` w ) = (TopSet ` W ) )
14	13, 3	syl6eqr 2145	. . . . 5 |- ( w = W -> (TopSet ` w ) = J )
15		fveq2 5340	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
16	15, 7	syl6eqr 2145	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
17	14, 16	oveq12d 5708	. . . 4 |- ( w = W -> ( (TopSet ` w ) ↾t ( Base ` w ) ) = ( J ↾t B ) )
18		df-topn 11807	. . . 4 |- TopOpen = ( w e. _V |-> ( (TopSet ` w ) ↾t ( Base ` w ) ) )
19	17, 18	fvmptg 5415	. . 3 |- ( ( W e. _V /\ ( J ↾t B ) e. _V ) -> ( TopOpen ` W ) = ( J ↾t B ) )
20	1, 12, 19	syl2anc 404	. 2 |- ( W e. V -> ( TopOpen ` W ) = ( J ↾t B ) )
21	20	eqcomd 2100	1 |- ( W e. V -> ( J ↾t B ) = ( TopOpen ` W ) )

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