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Theorem limsupcl 13350
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 3065 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 df-limsup 13348 . . . 4  |-  limsup  =  ( f  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( f "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
3 eqid 2400 . . . . . . 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 3657 . . . . . . . 8  |-  ( ( f " ( k [,) +oo ) )  i^i  RR* )  C_  RR*
5 supxrcl 11475 . . . . . . . 8  |-  ( ( ( f " (
k [,) +oo )
)  i^i  RR* )  C_  RR* 
->  sup ( ( ( f " ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
64, 5mp1i 13 . . . . . . 7  |-  ( k  e.  RR  ->  sup ( ( ( f
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
73, 6fmpti 5986 . . . . . 6  |-  ( k  e.  RR  |->  sup (
( ( f "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*
8 frn 5674 . . . . . 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 5 . . . . 5  |-  ran  (
k  e.  RR  |->  sup ( ( ( f
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  C_  RR*
10 infmxrcl 11477 . . . . 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 13 . . . 4  |-  ( f  e.  _V  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( f " ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) , 
RR* ,  `'  <  )  e.  RR* )
122, 11fmpti 5986 . . 3  |-  limsup : _V --> RR*
1312ffvelrni 5962 . 2  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
141, 13syl 17 1  |-  ( F  e.  V  ->  ( limsup `
 F )  e. 
RR* )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1840   _Vcvv 3056    i^i cin 3410    C_ wss 3411    |-> cmpt 4450   `'ccnv 4939   ran crn 4941   "cima 4943   -->wf 5519   ` cfv 5523  (class class class)co 6232   supcsup 7852   RRcr 9439   +oocpnf 9573   RR*cxr 9575    < clt 9576   [,)cico 11500   limsupclsp 13347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-limsup 13348
This theorem is referenced by:  limsuplt  13356  limsupbnd1  13359  caucvgrlem  13549  limsupre  36982
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