MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bcth2 Unicode version

Theorem bcth2 18752
Description: Baire's Category Theorem, version 2: If countably many closed sets cover  X, then one of them has an interior. (Contributed by Mario Carneiro, 10-Jan-2014.)
Hypothesis
Ref Expression
bcth.2  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
bcth2  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  E. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =/=  (/) )
Distinct variable groups:    D, k    k, J    k, M    k, X

Proof of Theorem bcth2
StepHypRef Expression
1 simpll 730 . 2  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  D  e.  ( CMet `  X )
)
2 simprl 732 . 2  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  M : NN
--> ( Clsd `  J
) )
3 cmetmet 18712 . . . . . . . 8  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
43ad2antrr 706 . . . . . . 7  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  D  e.  ( Met `  X ) )
5 metxmet 17899 . . . . . . 7  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
6 bcth.2 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
76mopntopon 17985 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  J  e.  (TopOn `  X )
)
84, 5, 73syl 18 . . . . . 6  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  J  e.  (TopOn `  X ) )
9 topontop 16664 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
108, 9syl 15 . . . . 5  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  J  e.  Top )
11 simprr 733 . . . . . . 7  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  U. ran  M  =  X )
12 toponmax 16666 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
138, 12syl 15 . . . . . . 7  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  X  e.  J )
1411, 13eqeltrd 2357 . . . . . 6  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  U. ran  M  e.  J )
15 elssuni 3855 . . . . . 6  |-  ( U. ran  M  e.  J  ->  U. ran  M  C_  U. J
)
1614, 15syl 15 . . . . 5  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  U. ran  M  C_ 
U. J )
17 isopn3i 16819 . . . . . 6  |-  ( ( J  e.  Top  /\  U.
ran  M  e.  J
)  ->  ( ( int `  J ) `  U. ran  M )  = 
U. ran  M )
1817adantlr 695 . . . . 5  |-  ( ( ( J  e.  Top  /\ 
U. ran  M  C_  U. J
)  /\  U. ran  M  e.  J )  ->  (
( int `  J
) `  U. ran  M
)  =  U. ran  M )
1910, 16, 14, 18syl21anc 1181 . . . 4  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  ( ( int `  J ) `  U. ran  M )  = 
U. ran  M )
2019, 11eqtrd 2315 . . 3  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  ( ( int `  J ) `  U. ran  M )  =  X )
21 simplr 731 . . 3  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  X  =/=  (/) )
2220, 21eqnetrd 2464 . 2  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  ( ( int `  J ) `  U. ran  M )  =/=  (/) )
236bcth 18751 . 2  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
)  /\  ( ( int `  J ) `  U. ran  M )  =/=  (/) )  ->  E. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =/=  (/) )
241, 2, 22, 23syl3anc 1182 1  |-  ( ( ( D  e.  (
CMet `  X )  /\  X  =/=  (/) )  /\  ( M : NN --> ( Clsd `  J )  /\  U. ran  M  =  X ) )  ->  E. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152   (/)c0 3455   U.cuni 3827   ran crn 4690   -->wf 5251   ` cfv 5255   NNcn 9746   * Metcxmt 16369   Metcme 16370   MetOpencmopn 16372   Topctop 16631  TopOnctopon 16632   Clsdccld 16753   intcnt 16754   CMetcms 18680
This theorem is referenced by:  ubthlem1  21449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-dc 8072  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lm 16959  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-cfil 18681  df-cau 18682  df-cmet 18683
  Copyright terms: Public domain W3C validator