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Theorem limsupcl 13072
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 3087 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 df-limsup 13070 . . . 4  |-  limsup  =  ( f  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( f "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
3 eqid 2454 . . . . . . 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 3682 . . . . . . . 8  |-  ( ( f " ( k [,) +oo ) )  i^i  RR* )  C_  RR*
5 supxrcl 11391 . . . . . . . 8  |-  ( ( ( f " (
k [,) +oo )
)  i^i  RR* )  C_  RR* 
->  sup ( ( ( f " ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
64, 5mp1i 12 . . . . . . 7  |-  ( k  e.  RR  ->  sup ( ( ( f
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
73, 6fmpti 5978 . . . . . 6  |-  ( k  e.  RR  |->  sup (
( ( f "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*
8 frn 5676 . . . . . 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 11393 . . . . 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 12 . . . 4  |-  ( f  e.  _V  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( f " ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) , 
RR* ,  `'  <  )  e.  RR* )
122, 11fmpti 5978 . . 3  |-  limsup : _V --> RR*
1312ffvelrni 5954 . 2  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
141, 13syl 16 1  |-  ( F  e.  V  ->  ( limsup `
 F )  e. 
RR* )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   _Vcvv 3078    i^i cin 3438    C_ wss 3439    |-> cmpt 4461   `'ccnv 4950   ran crn 4952   "cima 4954   -->wf 5525   ` cfv 5529  (class class class)co 6203   supcsup 7804   RRcr 9395   +oocpnf 9529   RR*cxr 9531    < clt 9532   [,)cico 11416   limsupclsp 13069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-limsup 13070
This theorem is referenced by:  limsuplt  13078  limsupbnd1  13081  caucvgrlem  13271
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