Theorem setsmsbasg 14869

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 12808	. . . 4 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
4		1re 8053	. . . . . 6 |- 1 e. RR
5		1lt9 9223	. . . . . 6 |- 1 < 9
6	4, 5	ltneii 8151	. . . . 5 |- 1 =/= 9
7		basendx 12806	. . . . . 6 |- ( Base ` ndx ) = 1
8		tsetndx 12936	. . . . . 6 |- (TopSet ` ndx ) = 9
9	7, 8	neeq12i 2392	. . . . 5 |- ( ( Base ` ndx ) =/= (TopSet ` ndx ) <-> 1 =/= 9 )
10	6, 9	mpbir 146	. . . 4 |- ( Base ` ndx ) =/= (TopSet ` ndx )
11		9nn 9187	. . . . 5 |- 9 e. NN
12	8, 11	eqeltri 2277	. . . 4 |- (TopSet ` ndx ) e. NN
13	3, 10, 12	setsslnid 12803	. . 3 |- ( ( M e. V /\ ( MetOpen ` D ) e. W ) -> ( Base ` M ) = ( Base ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
14	1, 2, 13	syl2anc 411	. 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 5574	. 2 |- ( ph -> ( Base ` K ) = ( Base ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
18	14, 15, 17	3eqtr4d 2247	1 |- ( ph -> X = ( Base ` K ) )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   =/= wne 2375   <.cop 3635   X. cxp 4671   |` cres 4675   ` cfv 5268  (class class class)co 5934   1c1 7908   NNcn 9018   9c9 9076   ndxcnx 12748   sSet csts 12749   Basecbs 12751  TopSetcts 12834   distcds 12837   MetOpencmopn 14221

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-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-addcom 8007  ax-addass 8009  ax-i2m1 8012  ax-0lt1 8013  ax-0id 8015  ax-rnegex 8016  ax-pre-ltirr 8019  ax-pre-lttrn 8021  ax-pre-ltadd 8023

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-iota 5229  df-fun 5270  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-pnf 8091  df-mnf 8092  df-ltxr 8094  df-inn 9019  df-2 9077  df-3 9078  df-4 9079  df-5 9080  df-6 9081  df-7 9082  df-8 9083  df-9 9084  df-ndx 12754  df-slot 12755  df-base 12757  df-sets 12758  df-tset 12847

This theorem is referenced by: (None)