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Theorem lmff 17029
Description: If  F converges, there is some upper integer set on which  F is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
Hypotheses
Ref Expression
lmff.1  |-  Z  =  ( ZZ>= `  M )
lmff.3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
lmff.4  |-  ( ph  ->  M  e.  ZZ )
lmff.5  |-  ( ph  ->  F  e.  dom  ( ~~> t `  J )
)
Assertion
Ref Expression
lmff  |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
Distinct variable groups:    j, F    j, J    j, M    ph, j    j, X    j, Z

Proof of Theorem lmff
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmff.5 . . . . . 6  |-  ( ph  ->  F  e.  dom  ( ~~> t `  J )
)
2 eldm2g 4875 . . . . . . 7  |-  ( F  e.  dom  ( ~~> t `  J )  ->  ( F  e.  dom  ( ~~> t `  J )  <->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) ) )
32ibi 232 . . . . . 6  |-  ( F  e.  dom  ( ~~> t `  J )  ->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) )
41, 3syl 15 . . . . 5  |-  ( ph  ->  E. y <. F , 
y >.  e.  ( ~~> t `  J ) )
5 df-br 4024 . . . . . 6  |-  ( F ( ~~> t `  J
) y  <->  <. F , 
y >.  e.  ( ~~> t `  J ) )
65exbii 1569 . . . . 5  |-  ( E. y  F ( ~~> t `  J ) y  <->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) )
74, 6sylibr 203 . . . 4  |-  ( ph  ->  E. y  F ( ~~> t `  J ) y )
8 lmff.3 . . . . . . . . 9  |-  ( ph  ->  J  e.  (TopOn `  X ) )
9 toponmax 16666 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
108, 9syl 15 . . . . . . . 8  |-  ( ph  ->  X  e.  J )
1110adantr 451 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  X  e.  J )
128lmbr 16988 . . . . . . . . 9  |-  ( ph  ->  ( F ( ~~> t `  J ) y  <->  ( F  e.  ( X  ^pm  CC )  /\  y  e.  X  /\  A. j  e.  J  ( y  e.  j  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j ) ) ) )
1312biimpa 470 . . . . . . . 8  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  ( F  e.  ( X  ^pm  CC )  /\  y  e.  X  /\  A. j  e.  J  ( y  e.  j  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j ) ) )
1413simp3d 969 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  A. j  e.  J  ( y  e.  j  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j ) )
15 lmcl 17025 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) y )  -> 
y  e.  X )
168, 15sylan 457 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  y  e.  X )
17 eleq2 2344 . . . . . . . . 9  |-  ( j  =  X  ->  (
y  e.  j  <->  y  e.  X ) )
18 feq3 5377 . . . . . . . . . 10  |-  ( j  =  X  ->  (
( F  |`  x
) : x --> j  <->  ( F  |`  x ) : x --> X ) )
1918rexbidv 2564 . . . . . . . . 9  |-  ( j  =  X  ->  ( E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j  <->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X ) )
2017, 19imbi12d 311 . . . . . . . 8  |-  ( j  =  X  ->  (
( y  e.  j  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j )  <-> 
( y  e.  X  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X ) ) )
2120rspcv 2880 . . . . . . 7  |-  ( X  e.  J  ->  ( A. j  e.  J  ( y  e.  j  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j )  ->  ( y  e.  X  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X ) ) )
2211, 14, 16, 21syl3c 57 . . . . . 6  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X )
2322ex 423 . . . . 5  |-  ( ph  ->  ( F ( ~~> t `  J ) y  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X ) )
2423exlimdv 1664 . . . 4  |-  ( ph  ->  ( E. y  F ( ~~> t `  J
) y  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X ) )
257, 24mpd 14 . . 3  |-  ( ph  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X )
26 uzf 10233 . . . 4  |-  ZZ>= : ZZ --> ~P ZZ
27 ffn 5389 . . . 4  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
28 reseq2 4950 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  ( F  |`  x )  =  ( F  |`  ( ZZ>= `  j ) ) )
29 id 19 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  x  =  ( ZZ>= `  j )
)
3028, 29feq12d 5381 . . . . 5  |-  ( x  =  ( ZZ>= `  j
)  ->  ( ( F  |`  x ) : x --> X  <->  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X ) )
3130rexrn 5667 . . . 4  |-  ( ZZ>=  Fn  ZZ  ->  ( E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X  <->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )
3226, 27, 31mp2b 9 . . 3  |-  ( E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X  <->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X )
3325, 32sylib 188 . 2  |-  ( ph  ->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
34 lmff.4 . . . 4  |-  ( ph  ->  M  e.  ZZ )
35 lmff.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
3635rexuz3 11832 . . . 4  |-  ( M  e.  ZZ  ->  ( E. j  e.  Z  A. x  e.  ( ZZ>=
`  j ) ( x  e.  dom  F  /\  ( F `  x
)  e.  X )  <->  E. j  e.  ZZ  A. x  e.  ( ZZ>= `  j ) ( x  e.  dom  F  /\  ( F `  x )  e.  X ) ) )
3734, 36syl 15 . . 3  |-  ( ph  ->  ( E. j  e.  Z  A. x  e.  ( ZZ>= `  j )
( x  e.  dom  F  /\  ( F `  x )  e.  X
)  <->  E. j  e.  ZZ  A. x  e.  ( ZZ>= `  j ) ( x  e.  dom  F  /\  ( F `  x )  e.  X ) ) )
3813simp1d 967 . . . . . . . . 9  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  F  e.  ( X  ^pm  CC ) )
3938ex 423 . . . . . . . 8  |-  ( ph  ->  ( F ( ~~> t `  J ) y  ->  F  e.  ( X  ^pm  CC ) ) )
4039exlimdv 1664 . . . . . . 7  |-  ( ph  ->  ( E. y  F ( ~~> t `  J
) y  ->  F  e.  ( X  ^pm  CC ) ) )
417, 40mpd 14 . . . . . 6  |-  ( ph  ->  F  e.  ( X 
^pm  CC ) )
42 pmfun 6790 . . . . . 6  |-  ( F  e.  ( X  ^pm  CC )  ->  Fun  F )
4341, 42syl 15 . . . . 5  |-  ( ph  ->  Fun  F )
44 ffvresb 5690 . . . . 5  |-  ( Fun 
F  ->  ( ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X  <->  A. x  e.  ( ZZ>=
`  j ) ( x  e.  dom  F  /\  ( F `  x
)  e.  X ) ) )
4543, 44syl 15 . . . 4  |-  ( ph  ->  ( ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X  <->  A. x  e.  ( ZZ>= `  j )
( x  e.  dom  F  /\  ( F `  x )  e.  X
) ) )
4645rexbidv 2564 . . 3  |-  ( ph  ->  ( E. j  e.  Z  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X  <->  E. j  e.  Z  A. x  e.  ( ZZ>= `  j )
( x  e.  dom  F  /\  ( F `  x )  e.  X
) ) )
4745rexbidv 2564 . . 3  |-  ( ph  ->  ( E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X  <->  E. j  e.  ZZ  A. x  e.  ( ZZ>= `  j )
( x  e.  dom  F  /\  ( F `  x )  e.  X
) ) )
4837, 46, 473bitr4d 276 . 2  |-  ( ph  ->  ( E. j  e.  Z  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X  <->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X ) )
4933, 48mpbird 223 1  |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   ~Pcpw 3625   <.cop 3643   class class class wbr 4023   dom cdm 4689   ran crn 4690    |` cres 4691   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   ZZcz 10024   ZZ>=cuz 10230  TopOnctopon 16632   ~~> tclm 16956
This theorem is referenced by:  lmle  18727
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-neg 9040  df-z 10025  df-uz 10231  df-top 16636  df-topon 16639  df-lm 16959
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