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Theorem ulmf2 19765
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)
Assertion
Ref Expression
ulmf2  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  F : Z --> ( CC  ^m  S ) )

Proof of Theorem ulmf2
StepHypRef Expression
1 ulmpm 19764 . . . 4  |-  ( F ( ~~> u `  S
) G  ->  F  e.  ( ( CC  ^m  S )  ^pm  ZZ ) )
2 ovex 5885 . . . . . 6  |-  ( CC 
^m  S )  e. 
_V
3 zex 10035 . . . . . 6  |-  ZZ  e.  _V
42, 3elpm2 6801 . . . . 5  |-  ( F  e.  ( ( CC 
^m  S )  ^pm  ZZ )  <->  ( F : dom  F --> ( CC  ^m  S )  /\  dom  F 
C_  ZZ ) )
54simplbi 446 . . . 4  |-  ( F  e.  ( ( CC 
^m  S )  ^pm  ZZ )  ->  F : dom  F --> ( CC  ^m  S ) )
61, 5syl 15 . . 3  |-  ( F ( ~~> u `  S
) G  ->  F : dom  F --> ( CC 
^m  S ) )
76adantl 452 . 2  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  F : dom  F --> ( CC  ^m  S
) )
8 fndm 5345 . . . 4  |-  ( F  Fn  Z  ->  dom  F  =  Z )
98adantr 451 . . 3  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  dom  F  =  Z )
109feq2d 5382 . 2  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  ( F : dom  F --> ( CC  ^m  S )  <->  F : Z
--> ( CC  ^m  S
) ) )
117, 10mpbid 201 1  |-  ( ( F  Fn  Z  /\  F ( ~~> u `  S ) G )  ->  F : Z --> ( CC  ^m  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686    C_ wss 3154   class class class wbr 4025   dom cdm 4691    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860    ^m cmap 6774    ^pm cpm 6775   CCcc 8737   ZZcz 10026   ~~> uculm 19757
This theorem is referenced by:  ulmdvlem1  19779  ulmdvlem2  19780  ulmdvlem3  19781  mbfulm  19784  iblulm  19785  itgulm  19786  itgulm2  19787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-map 6776  df-pm 6777  df-neg 9042  df-z 10027  df-uz 10233  df-ulm 19758
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