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Theorem limsupcl 11963
Description: Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 7-May-2016.)
Assertion
Ref Expression
limsupcl  |-  ( F  e.  V  ->  ( limsup `
 F )  e. 
RR* )

Proof of Theorem limsupcl
Dummy variables  f 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 df-limsup 11961 . . . 4  |-  limsup  =  ( f  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( f "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
3 eqid 2296 . . . . . . 7  |-  ( k  e.  RR  |->  sup (
( ( f "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ( k  e.  RR  |->  sup (
( ( f "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
4 inss2 3403 . . . . . . . 8  |-  ( ( f " ( k [,)  +oo ) )  i^i  RR* )  C_  RR*
5 supxrcl 10649 . . . . . . . 8  |-  ( ( ( f " (
k [,)  +oo ) )  i^i  RR* )  C_  RR*  ->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
64, 5mp1i 11 . . . . . . 7  |-  ( k  e.  RR  ->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
73, 6fmpti 5699 . . . . . 6  |-  ( k  e.  RR  |->  sup (
( ( f "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*
8 frn 5411 . . . . . 6  |-  ( ( k  e.  RR  |->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*  ->  ran  ( k  e.  RR  |->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  C_  RR* )
97, 8ax-mp 8 . . . . 5  |-  ran  (
k  e.  RR  |->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  C_  RR*
10 infmxrcl 10651 . . . . 5  |-  ( ran  ( k  e.  RR  |->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  C_  RR*  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( f " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  )  e.  RR* )
119, 10mp1i 11 . . . 4  |-  ( f  e.  _V  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( f " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  )  e.  RR* )
122, 11fmpti 5699 . . 3  |-  limsup : _V --> RR*
1312ffvelrni 5680 . 2  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
141, 13syl 15 1  |-  ( F  e.  V  ->  ( limsup `
 F )  e. 
RR* )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165    e. cmpt 4093   `'ccnv 4704   ran crn 4706   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752    +oocpnf 8880   RR*cxr 8882    < clt 8883   [,)cico 10674   limsupclsp 11960
This theorem is referenced by:  limsuplt  11969  limsupbnd1  11972  caucvgrlem  12161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-limsup 11961
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