Theorem topnvalg 13116

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 2783	. . 3 |- ( W e. V -> W e. _V )
2		restfn 13108	. . . 4 |- ↾t Fn ( _V X. _V )
3		topnval.2	. . . . 5 |- J = (TopSet ` W )
4		tsetslid 13053	. . . . . 6 |- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
5	4	slotex 12892	. . . . 5 |- ( W e. V -> (TopSet ` W ) e. _V )
6	3, 5	eqeltrid 2292	. . . 4 |- ( W e. V -> J e. _V )
7		topnval.1	. . . . 5 |- B = ( Base ` W )
8		baseslid 12922	. . . . . 6 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
9	8	slotex 12892	. . . . 5 |- ( W e. V -> ( Base ` W ) e. _V )
10	7, 9	eqeltrid 2292	. . . 4 |- ( W e. V -> B e. _V )
11		fnovex 5979	. . . 4 |- ( ( ↾t Fn ( _V X. _V ) /\ J e. _V /\ B e. _V ) -> ( J ↾t B ) e. _V )
12	2, 6, 10, 11	mp3an2i 1355	. . 3 |- ( W e. V -> ( J ↾t B ) e. _V )
13		fveq2 5578	. . . . . 6 |- ( w = W -> (TopSet ` w ) = (TopSet ` W ) )
14	13, 3	eqtr4di 2256	. . . . 5 |- ( w = W -> (TopSet ` w ) = J )
15		fveq2 5578	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
16	15, 7	eqtr4di 2256	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
17	14, 16	oveq12d 5964	. . . 4 |- ( w = W -> ( (TopSet ` w ) ↾t ( Base ` w ) ) = ( J ↾t B ) )
18		df-topn 13107	. . . 4 |- TopOpen = ( w e. _V |-> ( (TopSet ` w ) ↾t ( Base ` w ) ) )
19	17, 18	fvmptg 5657	. . 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 2211	1 |- ( W e. V -> ( J ↾t B ) = ( TopOpen ` W ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   X. cxp 4674   Fn wfn 5267   ` cfv 5272  (class class class)co 5946   Basecbs 12865  TopSetcts 12948   ↾_t crest 13104   TopOpenctopn 13105

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-5 9100  df-6 9101  df-7 9102  df-8 9103  df-9 9104  df-ndx 12868  df-slot 12869  df-base 12871  df-tset 12961  df-rest 13106  df-topn 13107

This theorem is referenced by:  topnidg  13117  topnpropgd  13118  mgptopng  13724