Theorem topnpropgd 12758

Description: The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)

Hypotheses

Ref	Expression
topnpropd.1	|- ( ph -> ( Base ` K ) = ( Base ` L ) )
topnpropd.2	|- ( ph -> (TopSet ` K ) = (TopSet ` L ) )
topnpropgd.k	|- ( ph -> K e. V )
topnpropgd.l	|- ( ph -> L e. W )

Assertion

Ref	Expression
topnpropgd	|- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )

Proof of Theorem topnpropgd

Step	Hyp	Ref	Expression
1		topnpropd.2	. . 3 |- ( ph -> (TopSet ` K ) = (TopSet ` L ) )
2		topnpropd.1	. . 3 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
3	1, 2	oveq12d 5914	. 2 |- ( ph -> ( (TopSet ` K ) ↾t ( Base ` K ) ) = ( (TopSet ` L ) ↾t ( Base ` L ) ) )
4		topnpropgd.k	. . 3 |- ( ph -> K e. V )
5		eqid 2189	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		eqid 2189	. . . 4 |- (TopSet ` K ) = (TopSet ` K )
7	5, 6	topnvalg 12756	. . 3 |- ( K e. V -> ( (TopSet ` K ) ↾t ( Base ` K ) ) = ( TopOpen ` K ) )
8	4, 7	syl 14	. 2 |- ( ph -> ( (TopSet ` K ) ↾t ( Base ` K ) ) = ( TopOpen ` K ) )
9		topnpropgd.l	. . 3 |- ( ph -> L e. W )
10		eqid 2189	. . . 4 |- ( Base ` L ) = ( Base ` L )
11		eqid 2189	. . . 4 |- (TopSet ` L ) = (TopSet ` L )
12	10, 11	topnvalg 12756	. . 3 |- ( L e. W -> ( (TopSet ` L ) ↾t ( Base ` L ) ) = ( TopOpen ` L ) )
13	9, 12	syl 14	. 2 |- ( ph -> ( (TopSet ` L ) ↾t ( Base ` L ) ) = ( TopOpen ` L ) )
14	3, 8, 13	3eqtr3d 2230	1 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   ` cfv 5235  (class class class)co 5896   Basecbs 12512  TopSetcts 12595   ↾_t crest 12744   TopOpenctopn 12745

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7932  ax-resscn 7933  ax-1re 7935  ax-addrcl 7938

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-5 9011  df-6 9012  df-7 9013  df-8 9014  df-9 9015  df-ndx 12515  df-slot 12516  df-base 12518  df-tset 12608  df-rest 12746  df-topn 12747

This theorem is referenced by:  sratopng  13763