MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fncpn Unicode version

Theorem fncpn 19386
Description: The  C ^n object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
fncpn  |-  ( S 
C_  CC  ->  ( C ^n `  S )  Fn  NN0 )

Proof of Theorem fncpn
Dummy variables  f  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 5970 . . . 4  |-  ( CC 
^pm  S )  e. 
_V
21rabex 4246 . . 3  |-  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) }  e.  _V
3 eqid 2358 . . 3  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } )
42, 3fnmpti 5454 . 2  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } )  Fn  NN0
5 cpnfval 19385 . . 3  |-  ( S 
C_  CC  ->  ( C ^n `  S )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } ) )
65fneq1d 5417 . 2  |-  ( S 
C_  CC  ->  ( ( C ^n `  S
)  Fn  NN0  <->  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  D n f ) `  n )  e.  ( dom  f -cn->
CC ) } )  Fn  NN0 ) )
74, 6mpbiri 224 1  |-  ( S 
C_  CC  ->  ( C ^n `  S )  Fn  NN0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1710   {crab 2623    C_ wss 3228    e. cmpt 4158   dom cdm 4771    Fn wfn 5332   ` cfv 5337  (class class class)co 5945    ^pm cpm 6861   CCcc 8825   NN0cn0 10057   -cn->ccncf 18483    D ncdvn 19318   C ^nccpn 19319
This theorem is referenced by:  cpncn  19389  cpnres  19390  plycpn  19773  aalioulem3  19818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-i2m1 8895  ax-1ne0 8896  ax-rrecex 8899  ax-cnre 8900
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-recs 6475  df-rdg 6510  df-nn 9837  df-n0 10058  df-cpn 19323
  Copyright terms: Public domain W3C validator