Theorem topnpropgd 13466

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 6068	. 2 |- ( ph -> ( (TopSet ` K ) ↾t ( Base ` K ) ) = ( (TopSet ` L ) ↾t ( Base ` L ) ) )
4		topnpropgd.k	. . 3 |- ( ph -> K e. V )
5		eqid 2232	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		eqid 2232	. . . 4 |- (TopSet ` K ) = (TopSet ` K )
7	5, 6	topnvalg 13464	. . 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 2232	. . . 4 |- ( Base ` L ) = ( Base ` L )
11		eqid 2232	. . . 4 |- (TopSet ` L ) = (TopSet ` L )
12	10, 11	topnvalg 13464	. . 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 2273	1 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212  TopSetcts 13296   ↾_t crest 13452   TopOpenctopn 13453

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-ndx 13215  df-slot 13216  df-base 13218  df-tset 13309  df-rest 13454  df-topn 13455

This theorem is referenced by:  sratopng  14595