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Theorem limcresi 19235
Description: Any limit of  F is also a limit of the restriction of  F. (Contributed by Mario Carneiro, 28-Dec-2016.)
Assertion
Ref Expression
limcresi  |-  ( F lim
CC  B )  C_  ( ( F  |`  C ) lim CC  B )

Proof of Theorem limcresi
Dummy variables  v  u  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limcrcl 19224 . . . . . . 7  |-  ( x  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
21simp1d 967 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  F : dom  F --> CC )
31simp2d 968 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  dom  F 
C_  CC )
41simp3d 969 . . . . . 6  |-  ( x  e.  ( F lim CC  B )  ->  B  e.  CC )
5 eqid 2283 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
62, 3, 4, 5ellimc2 19227 . . . . 5  |-  ( x  e.  ( F lim CC  B )  ->  (
x  e.  ( F lim
CC  B )  <->  ( x  e.  CC  /\  A. u  e.  ( TopOpen ` fld ) ( x  e.  u  ->  E. v  e.  ( TopOpen ` fld ) ( B  e.  v  /\  ( F
" ( v  i^i  ( dom  F  \  { B } ) ) )  C_  u )
) ) ) )
76ibi 232 . . . 4  |-  ( x  e.  ( F lim CC  B )  ->  (
x  e.  CC  /\  A. u  e.  ( TopOpen ` fld )
( x  e.  u  ->  E. v  e.  (
TopOpen ` fld ) ( B  e.  v  /\  ( F
" ( v  i^i  ( dom  F  \  { B } ) ) )  C_  u )
) ) )
8 inss2 3390 . . . . . . . . . . . . 13  |-  ( v  i^i  ( ( dom 
F  i^i  C )  \  { B } ) )  C_  ( ( dom  F  i^i  C ) 
\  { B }
)
9 difss 3303 . . . . . . . . . . . . . 14  |-  ( ( dom  F  i^i  C
)  \  { B } )  C_  ( dom  F  i^i  C )
10 inss2 3390 . . . . . . . . . . . . . 14  |-  ( dom 
F  i^i  C )  C_  C
119, 10sstri 3188 . . . . . . . . . . . . 13  |-  ( ( dom  F  i^i  C
)  \  { B } )  C_  C
128, 11sstri 3188 . . . . . . . . . . . 12  |-  ( v  i^i  ( ( dom 
F  i^i  C )  \  { B } ) )  C_  C
13 resima2 4988 . . . . . . . . . . . 12  |-  ( ( v  i^i  ( ( dom  F  i^i  C
)  \  { B } ) )  C_  C  ->  ( ( F  |`  C ) " (
v  i^i  ( ( dom  F  i^i  C ) 
\  { B }
) ) )  =  ( F " (
v  i^i  ( ( dom  F  i^i  C ) 
\  { B }
) ) ) )
1412, 13ax-mp 8 . . . . . . . . . . 11  |-  ( ( F  |`  C ) " ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) )  =  ( F "
( v  i^i  (
( dom  F  i^i  C )  \  { B } ) ) )
15 inss1 3389 . . . . . . . . . . . . 13  |-  ( dom 
F  i^i  C )  C_ 
dom  F
16 ssdif 3311 . . . . . . . . . . . . 13  |-  ( ( dom  F  i^i  C
)  C_  dom  F  -> 
( ( dom  F  i^i  C )  \  { B } )  C_  ( dom  F  \  { B } ) )
1715, 16ax-mp 8 . . . . . . . . . . . 12  |-  ( ( dom  F  i^i  C
)  \  { B } )  C_  ( dom  F  \  { B } )
18 sslin 3395 . . . . . . . . . . . 12  |-  ( ( ( dom  F  i^i  C )  \  { B } )  C_  ( dom  F  \  { B } )  ->  (
v  i^i  ( ( dom  F  i^i  C ) 
\  { B }
) )  C_  (
v  i^i  ( dom  F 
\  { B }
) ) )
19 imass2 5049 . . . . . . . . . . . 12  |-  ( ( v  i^i  ( ( dom  F  i^i  C
)  \  { B } ) )  C_  ( v  i^i  ( dom  F  \  { B } ) )  -> 
( F " (
v  i^i  ( ( dom  F  i^i  C ) 
\  { B }
) ) )  C_  ( F " ( v  i^i  ( dom  F  \  { B } ) ) ) )
2017, 18, 19mp2b 9 . . . . . . . . . . 11  |-  ( F
" ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) ) 
C_  ( F "
( v  i^i  ( dom  F  \  { B } ) ) )
2114, 20eqsstri 3208 . . . . . . . . . 10  |-  ( ( F  |`  C ) " ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) ) 
C_  ( F "
( v  i^i  ( dom  F  \  { B } ) ) )
22 sstr 3187 . . . . . . . . . 10  |-  ( ( ( ( F  |`  C ) " (
v  i^i  ( ( dom  F  i^i  C ) 
\  { B }
) ) )  C_  ( F " ( v  i^i  ( dom  F  \  { B } ) ) )  /\  ( F " ( v  i^i  ( dom  F  \  { B } ) ) )  C_  u )  ->  ( ( F  |`  C ) " (
v  i^i  ( ( dom  F  i^i  C ) 
\  { B }
) ) )  C_  u )
2321, 22mpan 651 . . . . . . . . 9  |-  ( ( F " ( v  i^i  ( dom  F  \  { B } ) ) )  C_  u  ->  ( ( F  |`  C ) " (
v  i^i  ( ( dom  F  i^i  C ) 
\  { B }
) ) )  C_  u )
2423anim2i 552 . . . . . . . 8  |-  ( ( B  e.  v  /\  ( F " ( v  i^i  ( dom  F  \  { B } ) ) )  C_  u
)  ->  ( B  e.  v  /\  (
( F  |`  C )
" ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) ) 
C_  u ) )
2524reximi 2650 . . . . . . 7  |-  ( E. v  e.  ( TopOpen ` fld )
( B  e.  v  /\  ( F "
( v  i^i  ( dom  F  \  { B } ) ) ) 
C_  u )  ->  E. v  e.  ( TopOpen
` fld
) ( B  e.  v  /\  ( ( F  |`  C ) " ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) ) 
C_  u ) )
2625imim2i 13 . . . . . 6  |-  ( ( x  e.  u  ->  E. v  e.  ( TopOpen
` fld
) ( B  e.  v  /\  ( F
" ( v  i^i  ( dom  F  \  { B } ) ) )  C_  u )
)  ->  ( x  e.  u  ->  E. v  e.  ( TopOpen ` fld ) ( B  e.  v  /\  ( ( F  |`  C ) " ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) ) 
C_  u ) ) )
2726ralimi 2618 . . . . 5  |-  ( A. u  e.  ( TopOpen ` fld )
( x  e.  u  ->  E. v  e.  (
TopOpen ` fld ) ( B  e.  v  /\  ( F
" ( v  i^i  ( dom  F  \  { B } ) ) )  C_  u )
)  ->  A. u  e.  ( TopOpen ` fld ) ( x  e.  u  ->  E. v  e.  ( TopOpen ` fld ) ( B  e.  v  /\  ( ( F  |`  C ) " ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) ) 
C_  u ) ) )
2827anim2i 552 . . . 4  |-  ( ( x  e.  CC  /\  A. u  e.  ( TopOpen ` fld )
( x  e.  u  ->  E. v  e.  (
TopOpen ` fld ) ( B  e.  v  /\  ( F
" ( v  i^i  ( dom  F  \  { B } ) ) )  C_  u )
) )  ->  (
x  e.  CC  /\  A. u  e.  ( TopOpen ` fld )
( x  e.  u  ->  E. v  e.  (
TopOpen ` fld ) ( B  e.  v  /\  ( ( F  |`  C ) " ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) ) 
C_  u ) ) ) )
297, 28syl 15 . . 3  |-  ( x  e.  ( F lim CC  B )  ->  (
x  e.  CC  /\  A. u  e.  ( TopOpen ` fld )
( x  e.  u  ->  E. v  e.  (
TopOpen ` fld ) ( B  e.  v  /\  ( ( F  |`  C ) " ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) ) 
C_  u ) ) ) )
30 fresin 5410 . . . . 5  |-  ( F : dom  F --> CC  ->  ( F  |`  C ) : ( dom  F  i^i  C ) --> CC )
312, 30syl 15 . . . 4  |-  ( x  e.  ( F lim CC  B )  ->  ( F  |`  C ) : ( dom  F  i^i  C ) --> CC )
3215, 3syl5ss 3190 . . . 4  |-  ( x  e.  ( F lim CC  B )  ->  ( dom  F  i^i  C ) 
C_  CC )
3331, 32, 4, 5ellimc2 19227 . . 3  |-  ( x  e.  ( F lim CC  B )  ->  (
x  e.  ( ( F  |`  C ) lim CC  B )  <->  ( x  e.  CC  /\  A. u  e.  ( TopOpen ` fld ) ( x  e.  u  ->  E. v  e.  ( TopOpen ` fld ) ( B  e.  v  /\  ( ( F  |`  C ) " ( v  i^i  ( ( dom  F  i^i  C )  \  { B } ) ) ) 
C_  u ) ) ) ) )
3429, 33mpbird 223 . 2  |-  ( x  e.  ( F lim CC  B )  ->  x  e.  ( ( F  |`  C ) lim CC  B ) )
3534ssriv 3184 1  |-  ( F lim
CC  B )  C_  ( ( F  |`  C ) lim CC  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   dom cdm 4689    |` cres 4691   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   TopOpenctopn 13326  ℂfldccnfld 16377   lim CC climc 19212
This theorem is referenced by:  limciun  19244  dvres2lem  19260  dvidlem  19265  dvcnp2  19269  dvcobr  19295  dvcnvlem  19323  lhop1lem  19360  lhop2  19362  lhop  19363  taylthlem2  19753
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-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-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-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  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-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cnp 16958  df-xms 17885  df-ms 17886  df-limc 19216
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