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Theorem lmflf 19422
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 10854 . . . . . . . 8  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5549 . . . . . . . 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 10870 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
64, 5eqsstri 3376 . . . . . . 7  |-  Z  C_  ZZ
7 imaeq2 5155 . . . . . . . . 9  |-  ( y  =  ( ZZ>= `  j
)  ->  ( F " y )  =  ( F " ( ZZ>= `  j ) ) )
87sseq1d 3373 . . . . . . . 8  |-  ( y  =  ( ZZ>= `  j
)  ->  ( ( F " y )  C_  x 
<->  ( F " ( ZZ>=
`  j ) ) 
C_  x ) )
98rexima 5945 . . . . . . 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 667 . . . . . 6  |-  ( E. y  e.  ( ZZ>= " Z ) ( F
" y )  C_  x 
<->  E. j  e.  Z  ( F " ( ZZ>= `  j ) )  C_  x )
11 simpl3 988 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  F : Z --> X )
12 ffun 5551 . . . . . . . . 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 10871 . . . . . . . . . . 11  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
1514, 4eleq2s 2527 . . . . . . . . . 10  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  ( ZZ>= `  M )
)
1615adantl 463 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  ( ZZ>=
`  j )  C_  ( ZZ>= `  M )
)
17 fdm 5553 . . . . . . . . . . 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 2483 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  dom  F  =  ( ZZ>= `  M
) )
2016, 19sseqtr4d 3383 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  ( ZZ>=
`  j )  C_  dom  F )
21 funimass4 5732 . . . . . . . 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 656 . . . . . . 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 2724 . . . . . 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 2727 . . 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 698 . 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 983 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  J  e.  (TopOn `  X )
)
29 simp2 984 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  M  e.  ZZ )
30 simp3 985 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  F : Z --> X )
31 eqidd 2436 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
3228, 4, 29, 30, 31lmbrf 18708 . 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 19315 . . 3  |-  ( M  e.  ZZ  ->  ( ZZ>=
" Z )  e.  ( fBas `  Z
) )
34 lmflf.2 . . . 4  |-  L  =  ( Z filGen ( ZZ>= " Z ) )
3534flffbas 19412 . . 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 1247 . 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 960    = wceq 1364    e. wcel 1757   A.wral 2707   E.wrex 2708    C_ wss 3318   ~Pcpw 3850   class class class wbr 4282   dom cdm 4829   "cima 4832   Fun wfun 5402    Fn wfn 5403   -->wf 5404   ` cfv 5408  (class class class)co 6082   ZZcz 10636   ZZ>=cuz 10851   fBascfbas 17650   filGencfg 17651  TopOnctopon 18343   ~~> tclm 18674    fLimf cflf 19352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-om 6468  df-1st 6568  df-2nd 6569  df-recs 6820  df-rdg 6854  df-er 7091  df-map 7206  df-pm 7207  df-en 7301  df-dom 7302  df-sdom 7303  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-nn 10313  df-z 10637  df-uz 10852  df-rest 14346  df-fbas 17660  df-fg 17661  df-top 18347  df-topon 18350  df-ntr 18468  df-nei 18546  df-lm 18677  df-fil 19263  df-fm 19355  df-flim 19356  df-flf 19357
This theorem is referenced by:  cmetcaulem  20643
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