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Theorem lmff 12455
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 4742 . . . . . . 7  |-  ( F  e.  dom  ( ~~> t `  J )  ->  ( F  e.  dom  ( ~~> t `  J )  <->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) ) )
32ibi 175 . . . . . 6  |-  ( F  e.  dom  ( ~~> t `  J )  ->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) )
41, 3syl 14 . . . . 5  |-  ( ph  ->  E. y <. F , 
y >.  e.  ( ~~> t `  J ) )
5 df-br 3937 . . . . . 6  |-  ( F ( ~~> t `  J
) y  <->  <. F , 
y >.  e.  ( ~~> t `  J ) )
65exbii 1585 . . . . 5  |-  ( E. y  F ( ~~> t `  J ) y  <->  E. y <. F ,  y >.  e.  ( ~~> t `  J
) )
74, 6sylibr 133 . . . 4  |-  ( ph  ->  E. y  F ( ~~> t `  J ) y )
8 lmff.3 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
9 lmcl 12451 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  F
( ~~> t `  J
) y )  -> 
y  e.  X )
108, 9sylan 281 . . . . 5  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  y  e.  X )
11 eleq2 2204 . . . . . . 7  |-  ( j  =  X  ->  (
y  e.  j  <->  y  e.  X ) )
12 feq3 5264 . . . . . . . 8  |-  ( j  =  X  ->  (
( F  |`  x
) : x --> j  <->  ( F  |`  x ) : x --> X ) )
1312rexbidv 2439 . . . . . . 7  |-  ( j  =  X  ->  ( E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j  <->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X ) )
1411, 13imbi12d 233 . . . . . 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 ) ) )
158lmbr 12419 . . . . . . . 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 ) ) ) )
1615biimpa 294 . . . . . . 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 ) ) )
1716simp3d 996 . . . . . 6  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  A. j  e.  J  ( y  e.  j  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> j ) )
18 toponmax 12229 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
198, 18syl 14 . . . . . . 7  |-  ( ph  ->  X  e.  J )
2019adantr 274 . . . . . 6  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  X  e.  J )
2114, 17, 20rspcdva 2797 . . . . 5  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  (
y  e.  X  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X ) )
2210, 21mpd 13 . . . 4  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X )
237, 22exlimddv 1871 . . 3  |-  ( ph  ->  E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X )
24 uzf 9352 . . . 4  |-  ZZ>= : ZZ --> ~P ZZ
25 ffn 5279 . . . 4  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
26 reseq2 4821 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  ( F  |`  x )  =  ( F  |`  ( ZZ>= `  j ) ) )
27 id 19 . . . . . 6  |-  ( x  =  ( ZZ>= `  j
)  ->  x  =  ( ZZ>= `  j )
)
2826, 27feq12d 5269 . . . . 5  |-  ( x  =  ( ZZ>= `  j
)  ->  ( ( F  |`  x ) : x --> X  <->  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X ) )
2928rexrn 5564 . . . 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 8 . . 3  |-  ( E. x  e.  ran  ZZ>= ( F  |`  x ) : x --> X  <->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X )
3123, 30sylib 121 . 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 10793 . . . 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 14 . . 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 ) ) )
3616simp1d 994 . . . . . . 7  |-  ( (
ph  /\  F ( ~~> t `  J )
y )  ->  F  e.  ( X  ^pm  CC ) )
377, 36exlimddv 1871 . . . . . 6  |-  ( ph  ->  F  e.  ( X 
^pm  CC ) )
38 pmfun 6569 . . . . . 6  |-  ( F  e.  ( X  ^pm  CC )  ->  Fun  F )
3937, 38syl 14 . . . . 5  |-  ( ph  ->  Fun  F )
40 ffvresb 5590 . . . . 5  |-  ( Fun 
F  ->  ( ( F  |`  ( ZZ>= `  j
) ) : (
ZZ>= `  j ) --> X  <->  A. x  e.  ( ZZ>=
`  j ) ( x  e.  dom  F  /\  ( F `  x
)  e.  X ) ) )
4139, 40syl 14 . . . 4  |-  ( ph  ->  ( ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X  <->  A. x  e.  ( ZZ>= `  j )
( x  e.  dom  F  /\  ( F `  x )  e.  X
) ) )
4241rexbidv 2439 . . 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 2439 . . 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 219 . 2  |-  ( ph  ->  ( E. j  e.  Z  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X  <->  E. j  e.  ZZ  ( F  |`  ( ZZ>= `  j )
) : ( ZZ>= `  j ) --> X ) )
4531, 44mpbird 166 1  |-  ( ph  ->  E. j  e.  Z  ( F  |`  ( ZZ>= `  j ) ) : ( ZZ>= `  j ) --> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332   E.wex 1469    e. wcel 1481   A.wral 2417   E.wrex 2418   ~Pcpw 3514   <.cop 3534   class class class wbr 3936   dom cdm 4546   ran crn 4547    |` cres 4548   Fun wfun 5124    Fn wfn 5125   -->wf 5126   ` cfv 5130  (class class class)co 5781    ^pm cpm 6550   CCcc 7641   ZZcz 9077   ZZ>=cuz 9349  TopOnctopon 12214   ~~> tclm 12393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-pm 6552  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-inn 8744  df-n0 9001  df-z 9078  df-uz 9350  df-top 12202  df-topon 12215  df-lm 12396
This theorem is referenced by: (None)
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