Theorem topnvalg 12171

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 2700	. . 3 |- ( W e. V -> W e. _V )
2		restfn 12163	. . . 4 |- ↾t Fn ( _V X. _V )
3		topnval.2	. . . . 5 |- J = (TopSet ` W )
4		tsetslid 12148	. . . . . 6 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
5	4	slotex 12025	. . . . 5 |- ( W e. V -> (TopSet ` W ) e. _V )
6	3, 5	eqeltrid 2227	. . . 4 |- ( W e. V -> J e. _V )
7		topnval.1	. . . . 5 |- B = ( Base ` W )
8		baseslid 12054	. . . . . 6 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
9	8	slotex 12025	. . . . 5 |- ( W e. V -> ( Base ` W ) e. _V )
10	7, 9	eqeltrid 2227	. . . 4 |- ( W e. V -> B e. _V )
11		fnovex 5812	. . . 4 |- ( ( ↾t Fn ( _V X. _V ) /\ J e. _V /\ B e. _V ) -> ( J ↾t B ) e. _V )
12	2, 6, 10, 11	mp3an2i 1321	. . 3 |- ( W e. V -> ( J ↾t B ) e. _V )
13		fveq2 5429	. . . . . 6 |- ( w = W -> (TopSet ` w ) = (TopSet ` W ) )
14	13, 3	eqtr4di 2191	. . . . 5 |- ( w = W -> (TopSet ` w ) = J )
15		fveq2 5429	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
16	15, 7	eqtr4di 2191	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
17	14, 16	oveq12d 5800	. . . 4 |- ( w = W -> ( (TopSet ` w ) ↾t ( Base ` w ) ) = ( J ↾t B ) )
18		df-topn 12162	. . . 4 |- TopOpen = ( w e. _V |-> ( (TopSet ` w ) ↾t ( Base ` w ) ) )
19	17, 18	fvmptg 5505	. . 3 |- ( ( W e. _V /\ ( J ↾t B ) e. _V ) -> ( TopOpen ` W ) = ( J ↾t B ) )
20	1, 12, 19	syl2anc 409	. 2 |- ( W e. V -> ( TopOpen ` W ) = ( J ↾t B ) )
21	20	eqcomd 2146	1 |- ( W e. V -> ( J ↾t B ) = ( TopOpen ` W ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   _Vcvv 2689   X. cxp 4545   Fn wfn 5126   ` cfv 5131  (class class class)co 5782   Basecbs 11998  TopSetcts 12066   ↾_t crest 12159   TopOpenctopn 12160

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-cnex 7735  ax-resscn 7736  ax-1re 7738  ax-addrcl 7741

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-5 8806  df-6 8807  df-7 8808  df-8 8809  df-9 8810  df-ndx 12001  df-slot 12002  df-base 12004  df-tset 12079  df-rest 12161  df-topn 12162

This theorem is referenced by:  topnidg  12172  topnpropgd  12173