Theorem topnpropgd 12570

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 5860	. 2 |- ( ph -> ( (TopSet ` K ) ↾t ( Base ` K ) ) = ( (TopSet ` L ) ↾t ( Base ` L ) ) )
4		topnpropgd.k	. . 3 |- ( ph -> K e. V )
5		eqid 2165	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		eqid 2165	. . . 4 |- (TopSet ` K ) = (TopSet ` K )
7	5, 6	topnvalg 12568	. . 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 2165	. . . 4 |- ( Base ` L ) = ( Base ` L )
11		eqid 2165	. . . 4 |- (TopSet ` L ) = (TopSet ` L )
12	10, 11	topnvalg 12568	. . 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 2206	1 |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   ` cfv 5188  (class class class)co 5842   Basecbs 12394  TopSetcts 12463   ↾_t crest 12556   TopOpenctopn 12557

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-5 8919  df-6 8920  df-7 8921  df-8 8922  df-9 8923  df-ndx 12397  df-slot 12398  df-base 12400  df-tset 12476  df-rest 12558  df-topn 12559

This theorem is referenced by: (None)