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Theorem lmff 19668
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 5205 . . . . . . 7  |-  ( F  e.  dom  ( ~~> t `  J )  ->  ( F  e.  dom  ( ~~> t `  J )  <->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) ) )
32ibi 241 . . . . . 6  |-  ( F  e.  dom  ( ~~> t `  J )  ->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) )
41, 3syl 16 . . . . 5  |-  ( ph  ->  E. y <. F , 
y >.  e.  ( ~~> t `  J ) )
5 df-br 4454 . . . . . 6  |-  ( F ( ~~> t `  J
) y  <->  <. F , 
y >.  e.  ( ~~> t `  J ) )
65exbii 1644 . . . . 5  |-  ( E. y  F ( ~~> t `  J ) y  <->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) )
74, 6sylibr 212 . . . 4  |-  ( ph  ->  E. y  F ( ~~> t `  J ) y )
8 lmff.3 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  X ) )
9 toponmax 19296 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
108, 9syl 16 . . . . . 6  |-  ( ph  ->  X  e.  J )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  X  e.  J )
128lmbr 19625 . . . . . . 7  |-  ( 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 484 . . . . . 6  |-  ( (
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 1010 . . . . 5  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  A. j  e.  J  ( y  e.  j  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j ) )
15 lmcl 19664 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) y )  -> 
y  e.  X )
168, 15sylan 471 . . . . 5  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  y  e.  X )
17 eleq2 2540 . . . . . . 7  |-  ( j  =  X  ->  (
y  e.  j  <->  y  e.  X ) )
18 feq3 5721 . . . . . . . 8  |-  ( j  =  X  ->  (
( F  |`  x
) : x --> j  <->  ( F  |`  x ) : x --> X ) )
1918rexbidv 2978 . . . . . . 7  |-  ( j  =  X  ->  ( E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j  <->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X ) )
2017, 19imbi12d 320 . . . . . 6  |-  ( 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 3215 . . . . 5  |-  ( 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 61 . . . 4  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X )
237, 22exlimddv 1702 . . 3  |-  ( ph  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X )
24 uzf 11097 . . . 4  |-  ZZ>= : ZZ --> ~P ZZ
25 ffn 5737 . . . 4  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
26 reseq2 5274 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  ( F  |`  x )  =  ( F  |`  ( ZZ>= `  j ) ) )
27 id 22 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  x  =  ( ZZ>= `  j )
)
2826, 27feq12d 5726 . . . . 5  |-  ( x  =  ( ZZ>= `  j
)  ->  ( ( F  |`  x ) : x --> X  <->  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X ) )
2928rexrn 6034 . . . 4  |-  ( ZZ>=  Fn  ZZ  ->  ( E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X  <->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X ) )
3024, 25, 29mp2b 10 . . 3  |-  ( E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X  <->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X )
3123, 30sylib 196 . 2  |-  ( ph  ->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
32 lmff.4 . . . 4  |-  ( ph  ->  M  e.  ZZ )
33 lmff.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
3433rexuz3 13160 . . . 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 ) ) )
3532, 34syl 16 . . 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 ) ) )
3613simp1d 1008 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  F  e.  ( X  ^pm  CC ) )
377, 36exlimddv 1702 . . . . . 6  |-  ( ph  ->  F  e.  ( X 
^pm  CC ) )
38 pmfun 7450 . . . . . 6  |-  ( F  e.  ( X  ^pm  CC )  ->  Fun  F )
3937, 38syl 16 . . . . 5  |-  ( ph  ->  Fun  F )
40 ffvresb 6063 . . . . 5  |-  ( Fun 
F  ->  ( ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X  <->  A. x  e.  ( ZZ>=
`  j ) ( x  e.  dom  F  /\  ( F `  x
)  e.  X ) ) )
4139, 40syl 16 . . . 4  |-  ( ph  ->  ( ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X  <->  A. x  e.  ( ZZ>= `  j )
( x  e.  dom  F  /\  ( F `  x )  e.  X
) ) )
4241rexbidv 2978 . . 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
) ) )
4341rexbidv 2978 . . 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
) ) )
4435, 42, 433bitr4d 285 . 2  |-  ( ph  ->  ( E. j  e.  Z  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X  <->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X ) )
4531, 44mpbird 232 1  |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2817   E.wrex 2818   ~Pcpw 4016   <.cop 4039   class class class wbr 4453   dom cdm 5005   ran crn 5006    |` cres 5007   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^pm cpm 7433   CCcc 9502   ZZcz 10876   ZZ>=cuz 11094  TopOnctopon 19262   ~~> tclm 19593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-pre-lttri 9578  ax-pre-lttrn 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-neg 9820  df-z 10877  df-uz 11095  df-top 19266  df-topon 19269  df-lm 19596
This theorem is referenced by:  lmle  21606
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