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Theorem lmflf 20484
Description: The topological limit relation on functions can be written in terms of the filter limit along the filter generated by the upper integer sets. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
lmflf.1  |-  Z  =  ( ZZ>= `  M )
lmflf.2  |-  L  =  ( Z filGen ( ZZ>= " Z ) )
Assertion
Ref Expression
lmflf  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( F ( ~~> t `  J ) P  <->  P  e.  ( ( J  fLimf  L ) `  F ) ) )

Proof of Theorem lmflf
Dummy variables  j 
k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzf 11095 . . . . . . . 8  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5721 . . . . . . . 8  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
31, 2ax-mp 5 . . . . . . 7  |-  ZZ>=  Fn  ZZ
4 lmflf.1 . . . . . . . 8  |-  Z  =  ( ZZ>= `  M )
5 uzssz 11111 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
64, 5eqsstri 3519 . . . . . . 7  |-  Z  C_  ZZ
7 imaeq2 5323 . . . . . . . . 9  |-  ( y  =  ( ZZ>= `  j
)  ->  ( F " y )  =  ( F " ( ZZ>= `  j ) ) )
87sseq1d 3516 . . . . . . . 8  |-  ( y  =  ( ZZ>= `  j
)  ->  ( ( F " y )  C_  x 
<->  ( F " ( ZZ>=
`  j ) ) 
C_  x ) )
98rexima 6136 . . . . . . 7  |-  ( (
ZZ>=  Fn  ZZ  /\  Z  C_  ZZ )  ->  ( E. y  e.  ( ZZ>=
" Z ) ( F " y ) 
C_  x  <->  E. j  e.  Z  ( F " ( ZZ>= `  j )
)  C_  x )
)
103, 6, 9mp2an 672 . . . . . 6  |-  ( E. y  e.  ( ZZ>= " Z ) ( F
" y )  C_  x 
<->  E. j  e.  Z  ( F " ( ZZ>= `  j ) )  C_  x )
11 simpl3 1002 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  F : Z --> X )
12 ffun 5723 . . . . . . . . 9  |-  ( F : Z --> X  ->  Fun  F )
1311, 12syl 16 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  Fun  F )
14 uzss 11112 . . . . . . . . . . 11  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
1514, 4eleq2s 2551 . . . . . . . . . 10  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  ( ZZ>= `  M )
)
1615adantl 466 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  ( ZZ>=
`  j )  C_  ( ZZ>= `  M )
)
17 fdm 5725 . . . . . . . . . . 11  |-  ( F : Z --> X  ->  dom  F  =  Z )
1811, 17syl 16 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  dom  F  =  Z )
1918, 4syl6eq 2500 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  dom  F  =  ( ZZ>= `  M
) )
2016, 19sseqtr4d 3526 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  ( ZZ>=
`  j )  C_  dom  F )
21 funimass4 5909 . . . . . . . 8  |-  ( ( Fun  F  /\  ( ZZ>=
`  j )  C_  dom  F )  ->  (
( F " ( ZZ>=
`  j ) ) 
C_  x  <->  A. k  e.  ( ZZ>= `  j )
( F `  k
)  e.  x ) )
2213, 20, 21syl2anc 661 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  (
( F " ( ZZ>=
`  j ) ) 
C_  x  <->  A. k  e.  ( ZZ>= `  j )
( F `  k
)  e.  x ) )
2322rexbidva 2951 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( E. j  e.  Z  ( F " ( ZZ>= `  j ) )  C_  x 
<->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x ) )
2410, 23syl5rbb 258 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x  <->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) )
2524imbi2d 316 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  (
( P  e.  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x )  <->  ( P  e.  x  ->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) ) )
2625ralbidv 2882 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( A. x  e.  J  ( P  e.  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x )  <->  A. x  e.  J  ( P  e.  x  ->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) ) )
2726anbi2d 703 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  (
( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( F `  k )  e.  x ) )  <-> 
( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. y  e.  (
ZZ>= " Z ) ( F " y ) 
C_  x ) ) ) )
28 simp1 997 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  J  e.  (TopOn `  X )
)
29 simp2 998 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  M  e.  ZZ )
30 simp3 999 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  F : Z --> X )
31 eqidd 2444 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
3228, 4, 29, 30, 31lmbrf 19739 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( F ( ~~> t `  J ) P  <->  ( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( F `  k
)  e.  x ) ) ) )
334uzfbas 20377 . . 3  |-  ( M  e.  ZZ  ->  ( ZZ>=
" Z )  e.  ( fBas `  Z
) )
34 lmflf.2 . . . 4  |-  L  =  ( Z filGen ( ZZ>= " Z ) )
3534flffbas 20474 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( ZZ>=
" Z )  e.  ( fBas `  Z
)  /\  F : Z
--> X )  ->  ( P  e.  ( ( J  fLimf  L ) `  F )  <->  ( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) ) ) )
3633, 35syl3an2 1263 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( P  e.  ( ( J  fLimf  L ) `  F )  <->  ( P  e.  X  /\  A. x  e.  J  ( P  e.  x  ->  E. y  e.  ( ZZ>= " Z ) ( F " y ) 
C_  x ) ) ) )
3727, 32, 363bitr4d 285 1  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  ( F ( ~~> t `  J ) P  <->  P  e.  ( ( J  fLimf  L ) `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    C_ wss 3461   ~Pcpw 3997   class class class wbr 4437   dom cdm 4989   "cima 4992   Fun wfun 5572    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   ZZcz 10871   ZZ>=cuz 11092   fBascfbas 18385   filGencfg 18386  TopOnctopon 19373   ~~> tclm 19705    fLimf cflf 20414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-z 10872  df-uz 11093  df-rest 14802  df-fbas 18395  df-fg 18396  df-top 19377  df-topon 19380  df-ntr 19499  df-nei 19577  df-lm 19708  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419
This theorem is referenced by:  cmetcaulem  21705
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