Theorem topnpropgd 12593

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 5871	. 2 |- ( ph -> ( (TopSet ` K ) ↾t ( Base ` K ) ) = ( (TopSet ` L ) ↾t ( Base ` L ) ) )
4		topnpropgd.k	. . 3 |- ( ph -> K e. V )
5		eqid 2170	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		eqid 2170	. . . 4 |- (TopSet ` K ) = (TopSet ` K )
7	5, 6	topnvalg 12591	. . 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 2170	. . . 4 |- ( Base ` L ) = ( Base ` L )
11		eqid 2170	. . . 4 |- (TopSet ` L ) = (TopSet ` L )
12	10, 11	topnvalg 12591	. . 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 2211	1 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   ` cfv 5198  (class class class)co 5853   Basecbs 12416  TopSetcts 12486   ↾_t crest 12579   TopOpenctopn 12580

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-9 8944  df-ndx 12419  df-slot 12420  df-base 12422  df-tset 12499  df-rest 12581  df-topn 12582

This theorem is referenced by: (None)