Theorem setsmsbasg 14893

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 12831	. . . 4 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
4		1re 8070	. . . . . 6 |- 1 e. RR
5		1lt9 9240	. . . . . 6 |- 1 < 9
6	4, 5	ltneii 8168	. . . . 5 |- 1 =/= 9
7		basendx 12829	. . . . . 6 |- ( Base ` ndx ) = 1
8		tsetndx 12960	. . . . . 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 9204	. . . . 5 |- 9 e. NN
12	8, 11	eqeltri 2277	. . . 4 |- (TopSet ` ndx ) e. NN
13	3, 10, 12	setsslnid 12826	. . 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 5579	. 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 4672   |` cres 4676   ` cfv 5270  (class class class)co 5943   1c1 7925   NNcn 9035   9c9 9093   ndxcnx 12771   sSet csts 12772   Basecbs 12774  TopSetcts 12857   distcds 12860   MetOpencmopn 14245

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 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

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 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-5 9097  df-6 9098  df-7 9099  df-8 9100  df-9 9101  df-ndx 12777  df-slot 12778  df-base 12780  df-sets 12781  df-tset 12870

This theorem is referenced by: (None)