Theorem setsmsbasg 13273

Description: The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)

Hypotheses

Ref	Expression
setsms.x	|- ( ph -> X = ( Base ` M ) )
setsms.d	|- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
setsms.k	|- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
setsmsbasg.m	|- ( ph -> M e. V )
setsmsbasg.d	|- ( ph -> ( MetOpen ` D ) e. W )

Assertion

Ref	Expression
setsmsbasg	|- ( ph -> X = ( Base ` K ) )

Proof of Theorem setsmsbasg

Step	Hyp	Ref	Expression
1		setsmsbasg.m	. . 3 |- ( ph -> M e. V )
2		setsmsbasg.d	. . 3 |- ( ph -> ( MetOpen ` D ) e. W )
3		baseslid 12472	. . . 4 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
4		1re 7919	. . . . . 6 |- 1 e. RR
5		1lt9 9082	. . . . . 6 |- 1 < 9
6	4, 5	ltneii 8016	. . . . 5 |- 1 =/= 9
7		basendx 12470	. . . . . 6 |- ( Base ` ndx ) = 1
8		tsetndx 12566	. . . . . 6 |- (TopSet ` ndx ) = 9
9	7, 8	neeq12i 2357	. . . . 5 |- ( ( Base ` ndx ) =/= (TopSet ` ndx ) <-> 1 =/= 9 )
10	6, 9	mpbir 145	. . . 4 |- ( Base ` ndx ) =/= (TopSet ` ndx )
11		9nn 9046	. . . . 5 |- 9 e. NN
12	8, 11	eqeltri 2243	. . . 4 |- (TopSet ` ndx ) e. NN
13	3, 10, 12	setsslnid 12467	. . 3 |- ( ( M e. V /\ ( MetOpen ` D ) e. W ) -> ( Base ` M ) = ( Base ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
14	1, 2, 13	syl2anc 409	. 2 |- ( ph -> ( Base ` M ) = ( Base ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
15		setsms.x	. 2 |- ( ph -> X = ( Base ` M ) )
16		setsms.k	. . 3 |- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
17	16	fveq2d 5500	. 2 |- ( ph -> ( Base ` K ) = ( Base ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
18	14, 15, 17	3eqtr4d 2213	1 |- ( ph -> X = ( Base ` K ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   =/= wne 2340   <.cop 3586   X. cxp 4609   |` cres 4613   ` cfv 5198  (class class class)co 5853   1c1 7775   NNcn 8878   9c9 8936   ndxcnx 12413   sSet csts 12414   Basecbs 12416  TopSetcts 12486   distcds 12489   MetOpencmopn 12779

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-in1 609  ax-in2 610  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-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-pre-ltirr 7886  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  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-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  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-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  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-sets 12423  df-tset 12499

This theorem is referenced by: (None)