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Theorem mptelee 24523
Description: A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)
Assertion
Ref Expression
mptelee  |-  ( N  e.  NN  ->  (
( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  e.  ( EE `  N )  <->  A. k  e.  (
1 ... N ) ( A F B )  e.  RR ) )
Distinct variable group:    k, N
Allowed substitution hints:    A( k)    B( k)    F( k)

Proof of Theorem mptelee
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 elee 24522 . 2  |-  ( N  e.  NN  ->  (
( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  e.  ( EE `  N )  <-> 
( k  e.  ( 1 ... N ) 
|->  ( A F B ) ) : ( 1 ... N ) --> RR ) )
2 ovex 5883 . . . . 5  |-  ( A F B )  e. 
_V
3 eqid 2283 . . . . 5  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  =  ( k  e.  ( 1 ... N
)  |->  ( A F B ) )
42, 3fnmpti 5372 . . . 4  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  Fn  ( 1 ... N )
5 df-f 5259 . . . 4  |-  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) ) : ( 1 ... N ) --> RR  <->  ( ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  Fn  (
1 ... N )  /\  ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR ) )
64, 5mpbiran 884 . . 3  |-  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) ) : ( 1 ... N ) --> RR  <->  ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR )
73rnmpt 4925 . . . . 5  |-  ran  (
k  e.  ( 1 ... N )  |->  ( A F B ) )  =  { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }
87sseq1i 3202 . . . 4  |-  ( ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR  <->  { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }  C_  RR )
9 abss 3242 . . . . 5  |-  ( { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }  C_  RR  <->  A. a
( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR ) )
10 nfre1 2599 . . . . . . . . 9  |-  F/ k E. k  e.  ( 1 ... N ) a  =  ( A F B )
11 nfv 1605 . . . . . . . . 9  |-  F/ k  a  e.  RR
1210, 11nfim 1769 . . . . . . . 8  |-  F/ k ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR )
1312nfal 1766 . . . . . . 7  |-  F/ k A. a ( E. k  e.  ( 1 ... N ) a  =  ( A F B )  ->  a  e.  RR )
14 r19.23v 2659 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) ( a  =  ( A F B )  -> 
a  e.  RR )  <-> 
( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR ) )
1514albii 1553 . . . . . . . 8  |-  ( A. a A. k  e.  ( 1 ... N ) ( a  =  ( A F B )  ->  a  e.  RR ) 
<-> 
A. a ( E. k  e.  ( 1 ... N ) a  =  ( A F B )  ->  a  e.  RR ) )
16 ralcom4 2806 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) A. a ( a  =  ( A F B )  ->  a  e.  RR )  <->  A. a A. k  e.  ( 1 ... N
) ( a  =  ( A F B )  ->  a  e.  RR ) )
17 rsp 2603 . . . . . . . . . 10  |-  ( A. k  e.  ( 1 ... N ) A. a ( a  =  ( A F B )  ->  a  e.  RR )  ->  ( k  e.  ( 1 ... N )  ->  A. a
( a  =  ( A F B )  ->  a  e.  RR ) ) )
182clel2 2904 . . . . . . . . . 10  |-  ( ( A F B )  e.  RR  <->  A. a
( a  =  ( A F B )  ->  a  e.  RR ) )
1917, 18syl6ibr 218 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) A. a ( a  =  ( A F B )  ->  a  e.  RR )  ->  ( k  e.  ( 1 ... N )  ->  ( A F B )  e.  RR ) )
2016, 19sylbir 204 . . . . . . . 8  |-  ( A. a A. k  e.  ( 1 ... N ) ( a  =  ( A F B )  ->  a  e.  RR )  ->  ( k  e.  ( 1 ... N
)  ->  ( A F B )  e.  RR ) )
2115, 20sylbir 204 . . . . . . 7  |-  ( A. a ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR )  ->  ( k  e.  ( 1 ... N
)  ->  ( A F B )  e.  RR ) )
2213, 21ralrimi 2624 . . . . . 6  |-  ( A. a ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR )  ->  A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
23 nfra1 2593 . . . . . . . 8  |-  F/ k A. k  e.  ( 1 ... N ) ( A F B )  e.  RR
24 rsp 2603 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  (
k  e.  ( 1 ... N )  -> 
( A F B )  e.  RR ) )
25 eleq1a 2352 . . . . . . . . 9  |-  ( ( A F B )  e.  RR  ->  (
a  =  ( A F B )  -> 
a  e.  RR ) )
2624, 25syl6 29 . . . . . . . 8  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  (
k  e.  ( 1 ... N )  -> 
( a  =  ( A F B )  ->  a  e.  RR ) ) )
2723, 11, 26rexlimd 2664 . . . . . . 7  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  ( E. k  e.  (
1 ... N ) a  =  ( A F B )  ->  a  e.  RR ) )
2827alrimiv 1617 . . . . . 6  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  A. a
( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR ) )
2922, 28impbii 180 . . . . 5  |-  ( A. a ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR ) 
<-> 
A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
309, 29bitri 240 . . . 4  |-  ( { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }  C_  RR  <->  A. k  e.  ( 1 ... N
) ( A F B )  e.  RR )
318, 30bitri 240 . . 3  |-  ( ran  ( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  C_  RR  <->  A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
326, 31bitri 240 . 2  |-  ( ( k  e.  ( 1 ... N )  |->  ( A F B ) ) : ( 1 ... N ) --> RR  <->  A. k  e.  ( 1 ... N ) ( A F B )  e.  RR )
331, 32syl6bb 252 1  |-  ( N  e.  NN  ->  (
( k  e.  ( 1 ... N ) 
|->  ( A F B ) )  e.  ( EE `  N )  <->  A. k  e.  (
1 ... N ) ( A F B )  e.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152    e. cmpt 4077   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738   NNcn 9746   ...cfz 10782   EEcee 24516
This theorem is referenced by:  eleesub  24539  eleesubd  24540  axsegconlem1  24545  axsegconlem8  24552  axpasch  24569  axeuclidlem  24590  axcontlem2  24593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-ee 24519
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