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Theorem lmff 18904
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 5035 . . . . . . 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 4292 . . . . . 6  |-  ( F ( ~~> t `  J
) y  <->  <. F , 
y >.  e.  ( ~~> t `  J ) )
65exbii 1634 . . . . 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 18532 . . . . . . 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 18861 . . . . . . 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 1002 . . . . 5  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  A. j  e.  J  ( y  e.  j  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j ) )
15 lmcl 18900 . . . . . 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 2503 . . . . . . 7  |-  ( j  =  X  ->  (
y  e.  j  <->  y  e.  X ) )
18 feq3 5543 . . . . . . . 8  |-  ( j  =  X  ->  (
( F  |`  x
) : x --> j  <->  ( F  |`  x ) : x --> X ) )
1918rexbidv 2735 . . . . . . 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 3068 . . . . 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 1692 . . 3  |-  ( ph  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X )
24 uzf 10863 . . . 4  |-  ZZ>= : ZZ --> ~P ZZ
25 ffn 5558 . . . 4  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
26 reseq2 5104 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  ( F  |`  x )  =  ( F  |`  ( ZZ>= `  j ) ) )
27 id 22 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  x  =  ( ZZ>= `  j )
)
2826, 27feq12d 5547 . . . . 5  |-  ( x  =  ( ZZ>= `  j
)  ->  ( ( F  |`  x ) : x --> X  <->  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X ) )
2928rexrn 5844 . . . 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 12835 . . . 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 1000 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  F  e.  ( X  ^pm  CC ) )
377, 36exlimddv 1692 . . . . . 6  |-  ( ph  ->  F  e.  ( X 
^pm  CC ) )
38 pmfun 7231 . . . . . 6  |-  ( F  e.  ( X  ^pm  CC )  ->  Fun  F )
3937, 38syl 16 . . . . 5  |-  ( ph  ->  Fun  F )
40 ffvresb 5873 . . . . 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 2735 . . 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 2735 . . 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2714   E.wrex 2715   ~Pcpw 3859   <.cop 3882   class class class wbr 4291   dom cdm 4839   ran crn 4840    |` cres 4841   Fun wfun 5411    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090    ^pm cpm 7214   CCcc 9279   ZZcz 10645   ZZ>=cuz 10860  TopOnctopon 18498   ~~> tclm 18829
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-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-pre-lttri 9355  ax-pre-lttrn 9356
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7100  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-neg 9597  df-z 10646  df-uz 10861  df-top 18502  df-topon 18505  df-lm 18832
This theorem is referenced by:  lmle  20811
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