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Theorem lmff 17366
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 5067 . . . . . . 7  |-  ( F  e.  dom  ( ~~> t `  J )  ->  ( F  e.  dom  ( ~~> t `  J )  <->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) ) )
32ibi 234 . . . . . 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 4214 . . . . . 6  |-  ( F ( ~~> t `  J
) y  <->  <. F , 
y >.  e.  ( ~~> t `  J ) )
65exbii 1593 . . . . 5  |-  ( E. y  F ( ~~> t `  J ) y  <->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) )
74, 6sylibr 205 . . . 4  |-  ( ph  ->  E. y  F ( ~~> t `  J ) y )
8 lmff.3 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  X ) )
9 toponmax 16994 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
108, 9syl 16 . . . . . 6  |-  ( ph  ->  X  e.  J )
1110adantr 453 . . . . 5  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  X  e.  J )
128lmbr 17323 . . . . . . 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 472 . . . . . 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 972 . . . . 5  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  A. j  e.  J  ( y  e.  j  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j ) )
15 lmcl 17362 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) y )  -> 
y  e.  X )
168, 15sylan 459 . . . . 5  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  y  e.  X )
17 eleq2 2498 . . . . . . 7  |-  ( j  =  X  ->  (
y  e.  j  <->  y  e.  X ) )
18 feq3 5579 . . . . . . . 8  |-  ( j  =  X  ->  (
( F  |`  x
) : x --> j  <->  ( F  |`  x ) : x --> X ) )
1918rexbidv 2727 . . . . . . 7  |-  ( j  =  X  ->  ( E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j  <->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X ) )
2017, 19imbi12d 313 . . . . . 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 3049 . . . . 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 60 . . . 4  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X )
237, 22exlimddv 1649 . . 3  |-  ( ph  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X )
24 uzf 10492 . . . 4  |-  ZZ>= : ZZ --> ~P ZZ
25 ffn 5592 . . . 4  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
26 reseq2 5142 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  ( F  |`  x )  =  ( F  |`  ( ZZ>= `  j ) ) )
27 id 21 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  x  =  ( ZZ>= `  j )
)
2826, 27feq12d 5583 . . . . 5  |-  ( x  =  ( ZZ>= `  j
)  ->  ( ( F  |`  x ) : x --> X  <->  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X ) )
2928rexrn 5873 . . . 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 190 . 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 12153 . . . 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 970 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  F  e.  ( X  ^pm  CC ) )
377, 36exlimddv 1649 . . . . . 6  |-  ( ph  ->  F  e.  ( X 
^pm  CC ) )
38 pmfun 7037 . . . . . 6  |-  ( F  e.  ( X  ^pm  CC )  ->  Fun  F )
3937, 38syl 16 . . . . 5  |-  ( ph  ->  Fun  F )
40 ffvresb 5901 . . . . 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 2727 . . 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 2727 . . 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 278 . 2  |-  ( ph  ->  ( E. j  e.  Z  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X  <->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X ) )
4531, 44mpbird 225 1  |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707   ~Pcpw 3800   <.cop 3818   class class class wbr 4213   dom cdm 4879   ran crn 4880    |` cres 4881   Fun wfun 5449    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082    ^pm cpm 7020   CCcc 8989   ZZcz 10283   ZZ>=cuz 10489  TopOnctopon 16960   ~~> tclm 17291
This theorem is referenced by:  lmle  19255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-pre-lttri 9065  ax-pre-lttrn 9066
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-er 6906  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-neg 9295  df-z 10284  df-uz 10490  df-top 16964  df-topon 16967  df-lm 17294
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