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Theorem lmflf 17716
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 10249 . . . . . . . 8  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5405 . . . . . . . 8  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
31, 2ax-mp 8 . . . . . . 7  |-  ZZ>=  Fn  ZZ
4 lmflf.1 . . . . . . . 8  |-  Z  =  ( ZZ>= `  M )
5 uzssz 10263 . . . . . . . 8  |-  ( ZZ>= `  M )  C_  ZZ
64, 5eqsstri 3221 . . . . . . 7  |-  Z  C_  ZZ
7 imaeq2 5024 . . . . . . . . 9  |-  ( y  =  ( ZZ>= `  j
)  ->  ( F " y )  =  ( F " ( ZZ>= `  j ) ) )
87sseq1d 3218 . . . . . . . 8  |-  ( y  =  ( ZZ>= `  j
)  ->  ( ( F " y )  C_  x 
<->  ( F " ( ZZ>=
`  j ) ) 
C_  x ) )
98rexima 5773 . . . . . . 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 653 . . . . . 6  |-  ( E. y  e.  ( ZZ>= " Z ) ( F
" y )  C_  x 
<->  E. j  e.  Z  ( F " ( ZZ>= `  j ) )  C_  x )
11 simpl3 960 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  F : Z --> X )
12 ffun 5407 . . . . . . . . 9  |-  ( F : Z --> X  ->  Fun  F )
1311, 12syl 15 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  Fun  F )
14 uzss 10264 . . . . . . . . . . 11  |-  ( j  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  j )  C_  ( ZZ>=
`  M ) )
1514, 4eleq2s 2388 . . . . . . . . . 10  |-  ( j  e.  Z  ->  ( ZZ>=
`  j )  C_  ( ZZ>= `  M )
)
1615adantl 452 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  ( ZZ>=
`  j )  C_  ( ZZ>= `  M )
)
17 fdm 5409 . . . . . . . . . . 11  |-  ( F : Z --> X  ->  dom  F  =  Z )
1811, 17syl 15 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  dom  F  =  Z )
1918, 4syl6eq 2344 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  dom  F  =  ( ZZ>= `  M
) )
2016, 19sseqtr4d 3228 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  j  e.  Z )  ->  ( ZZ>=
`  j )  C_  dom  F )
21 funimass4 5589 . . . . . . . 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 642 . . . . . . 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 2573 . . . . . 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 249 . . . . 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 307 . . . 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 2576 . . 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 684 . 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 955 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  J  e.  (TopOn `  X )
)
29 simp2 956 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  M  e.  ZZ )
30 simp3 957 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  ->  F : Z --> X )
31 eqidd 2297 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  M  e.  ZZ  /\  F : Z
--> X )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
3228, 4, 29, 30, 31lmbrf 17006 . 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 17609 . . 3  |-  ( M  e.  ZZ  ->  ( ZZ>=
" Z )  e.  ( fBas `  Z
) )
34 lmflf.2 . . . 4  |-  L  =  ( Z filGen ( ZZ>= " Z ) )
3534flffbas 17706 . . 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 1216 . 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 276 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039   dom cdm 4705   "cima 4708   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   ZZcz 10040   ZZ>=cuz 10246  TopOnctopon 16648   ~~> tclm 16972   fBascfbas 17534   filGencfg 17535    fLimf cflf 17646
This theorem is referenced by:  cmetcaulem  18730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-z 10041  df-uz 10247  df-rest 13343  df-top 16652  df-topon 16655  df-ntr 16773  df-nei 16851  df-lm 16975  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651
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