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Theorem fncpn 22064
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 6300 . . . 4  |-  ( CC 
^pm  S )  e. 
_V
21rabex 4591 . . 3  |-  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) }  e.  _V
3 eqid 2460 . . 3  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } )
42, 3fnmpti 5700 . 2  |-  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } )  Fn  NN0
5 cpnfval 22063 . . 3  |-  ( S 
C_  CC  ->  ( C^n `  S )  =  ( n  e. 
NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } ) )
65fneq1d 5662 . 2  |-  ( S 
C_  CC  ->  ( ( C^n `  S
)  Fn  NN0  <->  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S
)  |  ( ( S  Dn f ) `  n )  e.  ( dom  f -cn->
CC ) } )  Fn  NN0 ) )
74, 6mpbiri 233 1  |-  ( S 
C_  CC  ->  ( C^n `  S )  Fn  NN0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1762   {crab 2811    C_ wss 3469    |-> cmpt 4498   dom cdm 4992    Fn wfn 5574   ` cfv 5579  (class class class)co 6275    ^pm cpm 7411   CCcc 9479   NN0cn0 10784   -cn->ccncf 21108    Dncdvn 21996   C^nccpn 21997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-om 6672  df-recs 7032  df-rdg 7066  df-nn 10526  df-n0 10785  df-cpn 22001
This theorem is referenced by:  cpncn  22067  cpnres  22068  plycpn  22412  aalioulem3  22457
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