Theorem topnval 16095

Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)

Hypotheses

Ref	Expression
topnval.1	|- B = ( Base ` W )
topnval.2	|- J = (TopSet ` W )

Assertion

Ref	Expression
topnval	|- ( J ↾t B ) = ( TopOpen ` W )

Proof of Theorem topnval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( w = W -> (TopSet ` w ) = (TopSet ` W ) )
2		topnval.2	. . . . . 6 |- J = (TopSet ` W )
3	1, 2	syl6eqr 2674	. . . . 5 |- ( w = W -> (TopSet ` w ) = J )
4		fveq2 6191	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
5		topnval.1	. . . . . 6 |- B = ( Base ` W )
6	4, 5	syl6eqr 2674	. . . . 5 |- ( w = W -> ( Base ` w ) = B )
7	3, 6	oveq12d 6668	. . . 4 |- ( w = W -> ( (TopSet ` w ) ↾t ( Base ` w ) ) = ( J ↾t B ) )
8		df-topn 16084	. . . 4 |- TopOpen = ( w e. _V |-> ( (TopSet ` w ) ↾t ( Base ` w ) ) )
9		ovex 6678	. . . 4 |- ( J ↾t B ) e. _V
10	7, 8, 9	fvmpt 6282	. . 3 |- ( W e. _V -> ( TopOpen ` W ) = ( J ↾t B ) )
11	10	eqcomd 2628	. 2 |- ( W e. _V -> ( J ↾t B ) = ( TopOpen ` W ) )
12		0rest 16090	. . 3 |- ( (/) ↾t B ) = (/)
13		fvprc 6185	. . . . 5 |- ( -. W e. _V -> (TopSet ` W ) = (/) )
14	2, 13	syl5eq 2668	. . . 4 |- ( -. W e. _V -> J = (/) )
15	14	oveq1d 6665	. . 3 |- ( -. W e. _V -> ( J ↾t B ) = ( (/) ↾t B ) )
16		fvprc 6185	. . 3 |- ( -. W e. _V -> ( TopOpen ` W ) = (/) )
17	12, 15, 16	3eqtr4a 2682	. 2 |- ( -. W e. _V -> ( J ↾t B ) = ( TopOpen ` W ) )
18	11, 17	pm2.61i 176	1 |- ( J ↾t B ) = ( TopOpen ` W )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Basecbs 15857  TopSetcts 15947   ↾_t crest 16081   TopOpenctopn 16082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-rest 16083  df-topn 16084

This theorem is referenced by:  topnid  16096  topnpropd  16097  oppgtopn  17783  symgtopn  17825  mgptopn  18498  resstopn  20990  prdstopn  21431  tuslem  22071  xrge0tsms  22637  om1opn  22836  xrge0tsmsd  29785  xrge0tmdOLD  29991