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Theorem mptelee 24595
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 24594 . 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 5899 . . . . 5  |-  ( A F B )  e. 
_V
3 eqid 2296 . . . . 5  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  =  ( k  e.  ( 1 ... N
)  |->  ( A F B ) )
42, 3fnmpti 5388 . . . 4  |-  ( k  e.  ( 1 ... N )  |->  ( A F B ) )  Fn  ( 1 ... N )
5 df-f 5275 . . . 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 4941 . . . . 5  |-  ran  (
k  e.  ( 1 ... N )  |->  ( A F B ) )  =  { a  |  E. k  e.  ( 1 ... N
) a  =  ( A F B ) }
87sseq1i 3215 . . . 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 3255 . . . . 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 2612 . . . . . . . . 9  |-  F/ k E. k  e.  ( 1 ... N ) a  =  ( A F B )
11 nfv 1609 . . . . . . . . 9  |-  F/ k  a  e.  RR
1210, 11nfim 1781 . . . . . . . 8  |-  F/ k ( E. k  e.  ( 1 ... N
) a  =  ( A F B )  ->  a  e.  RR )
1312nfal 1778 . . . . . . 7  |-  F/ k A. a ( E. k  e.  ( 1 ... N ) a  =  ( A F B )  ->  a  e.  RR )
14 r19.23v 2672 . . . . . . . . 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 1556 . . . . . . . 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 2819 . . . . . . . . 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 2616 . . . . . . . . . 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 2917 . . . . . . . . . 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 2637 . . . . . 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 2606 . . . . . . . 8  |-  F/ k A. k  e.  ( 1 ... N ) ( A F B )  e.  RR
24 rsp 2616 . . . . . . . . 9  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  (
k  e.  ( 1 ... N )  -> 
( A F B )  e.  RR ) )
25 eleq1a 2365 . . . . . . . . 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 2677 . . . . . . 7  |-  ( A. k  e.  ( 1 ... N ) ( A F B )  e.  RR  ->  ( E. k  e.  (
1 ... N ) a  =  ( A F B )  ->  a  e.  RR ) )
2827alrimiv 1621 . . . . . 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 1530    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    C_ wss 3165    e. cmpt 4093   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754   NNcn 9762   ...cfz 10798   EEcee 24588
This theorem is referenced by:  eleesub  24611  eleesubd  24612  axsegconlem1  24617  axsegconlem8  24624  axpasch  24641  axeuclidlem  24662  axcontlem2  24665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-ee 24591
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