Theorem topnpropgd 13666

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 6106	. 2 |- ( ph -> ( (TopSet ` K ) ↾t ( Base ` K ) ) = ( (TopSet ` L ) ↾t ( Base ` L ) ) )
4		topnpropgd.k	. . 3 |- ( ph -> K e. V )
5		eqid 2238	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		eqid 2238	. . . 4 |- (TopSet ` K ) = (TopSet ` K )
7	5, 6	topnvalg 13664	. . 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 2238	. . . 4 |- ( Base ` L ) = ( Base ` L )
11		eqid 2238	. . . 4 |- (TopSet ` L ) = (TopSet ` L )
12	10, 11	topnvalg 13664	. . 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 2279	1 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   Basecbs 13410  TopSetcts 13496   ↾_t crest 13652   TopOpenctopn 13653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-ndx 13413  df-slot 13414  df-base 13416  df-tset 13509  df-rest 13654  df-topn 13655

This theorem is used by:  sratopng  14893