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Theorem lmflf 19600
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 10885 . . . . . . . 8  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5580 . . . . . . . 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 10901 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
64, 5eqsstri 3407 . . . . . . 7  |-  Z  C_  ZZ
7 imaeq2 5186 . . . . . . . . 9  |-  ( y  =  ( ZZ>= `  j
)  ->  ( F " y )  =  ( F " ( ZZ>= `  j ) ) )
87sseq1d 3404 . . . . . . . 8  |-  ( y  =  ( ZZ>= `  j
)  ->  ( ( F " y )  C_  x 
<->  ( F " ( ZZ>=
`  j ) ) 
C_  x ) )
98rexima 5977 . . . . . . 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 993 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  F : Z --> X )
12 ffun 5582 . . . . . . . . 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 10902 . . . . . . . . . . 11  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
1514, 4eleq2s 2535 . . . . . . . . . 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 5584 . . . . . . . . . . 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 2491 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  dom  F  =  ( ZZ>= `  M
) )
2016, 19sseqtr4d 3414 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  ( ZZ>=
`  j )  C_  dom  F )
21 funimass4 5763 . . . . . . . 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 2753 . . . . . 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 2756 . . 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 988 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  J  e.  (TopOn `  X )
)
29 simp2 989 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  M  e.  ZZ )
30 simp3 990 . . 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 18886 . 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 19493 . . 3  |-  ( M  e.  ZZ  ->  ( ZZ>=
" Z )  e.  ( fBas `  Z
) )
34 lmflf.2 . . . 4  |-  L  =  ( Z filGen ( ZZ>= " Z ) )
3534flffbas 19590 . . 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 1252 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737    C_ wss 3349   ~Pcpw 3881   class class class wbr 4313   dom cdm 4861   "cima 4864   Fun wfun 5433    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   ZZcz 10667   ZZ>=cuz 10882   fBascfbas 17826   filGencfg 17827  TopOnctopon 18521   ~~> tclm 18852    fLimf cflf 19530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-z 10668  df-uz 10883  df-rest 14382  df-fbas 17836  df-fg 17837  df-top 18525  df-topon 18528  df-ntr 18646  df-nei 18724  df-lm 18855  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535
This theorem is referenced by:  cmetcaulem  20821
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