Theorem setsmsbasg 15026

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 12964	. . . 4 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
4		1re 8091	. . . . . 6 |- 1 e. RR
5		1lt9 9261	. . . . . 6 |- 1 < 9
6	4, 5	ltneii 8189	. . . . 5 |- 1 =/= 9
7		basendx 12962	. . . . . 6 |- ( Base ` ndx ) = 1
8		tsetndx 13093	. . . . . 6 |- (TopSet ` ndx ) = 9
9	7, 8	neeq12i 2394	. . . . 5 |- ( ( Base ` ndx ) =/= (TopSet ` ndx ) <-> 1 =/= 9 )
10	6, 9	mpbir 146	. . . 4 |- ( Base ` ndx ) =/= (TopSet ` ndx )
11		9nn 9225	. . . . 5 |- 9 e. NN
12	8, 11	eqeltri 2279	. . . 4 |- (TopSet ` ndx ) e. NN
13	3, 10, 12	setsslnid 12959	. . 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 5593	. 2 |- ( ph -> ( Base ` K ) = ( Base ` ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
18	14, 15, 17	3eqtr4d 2249	1 |- ( ph -> X = ( Base ` K ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   =/= wne 2377   <.cop 3641   X. cxp 4681   |` cres 4685   ` cfv 5280  (class class class)co 5957   1c1 7946   NNcn 9056   9c9 9114   ndxcnx 12904   sSet csts 12905   Basecbs 12907  TopSetcts 12990   distcds 12993   MetOpencmopn 14378

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-iota 5241  df-fun 5282  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-9 9122  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-tset 13003

This theorem is referenced by: (None)