Theorem mptelee 25775

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.

Step	Hyp	Ref	Expression
1		elee 25774	. 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 6678	. . . . 5 |- ( A F B ) e. _V
3		eqid 2622	. . . . 5 |- ( k e. ( 1 ... N ) |-> ( A F B ) ) = ( k e. ( 1 ... N ) |-> ( A F B ) )
4	2, 3	fnmpti 6022	. . . 4 |- ( k e. ( 1 ... N ) |-> ( A F B ) ) Fn ( 1 ... N )
5		df-f 5892	. . . 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 ) )
6	4, 5	mpbiran 953	. . 3 |- ( ( k e. ( 1 ... N ) |-> ( A F B ) ) : ( 1 ... N ) --> RR <-> ran ( k e. ( 1 ... N ) |-> ( A F B ) ) C_ RR )
7	3	rnmpt 5371	. . . . 5 |- ran ( k e. ( 1 ... N ) |-> ( A F B ) ) = { a | E. k e. ( 1 ... N ) a = ( A F B ) }
8	7	sseq1i 3629	. . . 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 3671	. . . . 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 3005	. . . . . . . . 9 |- F/ k E. k e. ( 1 ... N ) a = ( A F B )
11		nfv 1843	. . . . . . . . 9 |- F/ k a e. RR
12	10, 11	nfim 1825	. . . . . . . 8 |- F/ k ( E. k e. ( 1 ... N ) a = ( A F B ) -> a e. RR )
13	12	nfal 2153	. . . . . . 7 |- F/ k A. a ( E. k e. ( 1 ... N ) a = ( A F B ) -> a e. RR )
14		r19.23v 3023	. . . . . . . . 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 ) )
15	14	albii 1747	. . . . . . . 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 3224	. . . . . . . . 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 2929	. . . . . . . . . 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 ) ) )
18	2	clel2 3339	. . . . . . . . . 10 |- ( ( A F B ) e. RR <-> A. a ( a = ( A F B ) -> a e. RR ) )
19	17, 18	syl6ibr 242	. . . . . . . . 9 |- ( A. k e. ( 1 ... N ) A. a ( a = ( A F B ) -> a e. RR ) -> ( k e. ( 1 ... N ) -> ( A F B ) e. RR ) )
20	16, 19	sylbir 225	. . . . . . . 8 |- ( A. a A. k e. ( 1 ... N ) ( a = ( A F B ) -> a e. RR ) -> ( k e. ( 1 ... N ) -> ( A F B ) e. RR ) )
21	15, 20	sylbir 225	. . . . . . 7 |- ( A. a ( E. k e. ( 1 ... N ) a = ( A F B ) -> a e. RR ) -> ( k e. ( 1 ... N ) -> ( A F B ) e. RR ) )
22	13, 21	ralrimi 2957	. . . . . 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 2941	. . . . . . . 8 |- F/ k A. k e. ( 1 ... N ) ( A F B ) e. RR
24		rsp 2929	. . . . . . . . 9 |- ( A. k e. ( 1 ... N ) ( A F B ) e. RR -> ( k e. ( 1 ... N ) -> ( A F B ) e. RR ) )
25		eleq1a 2696	. . . . . . . . 9 |- ( ( A F B ) e. RR -> ( a = ( A F B ) -> a e. RR ) )
26	24, 25	syl6 35	. . . . . . . 8 |- ( A. k e. ( 1 ... N ) ( A F B ) e. RR -> ( k e. ( 1 ... N ) -> ( a = ( A F B ) -> a e. RR ) ) )
27	23, 11, 26	rexlimd 3026	. . . . . . 7 |- ( A. k e. ( 1 ... N ) ( A F B ) e. RR -> ( E. k e. ( 1 ... N ) a = ( A F B ) -> a e. RR ) )
28	27	alrimiv 1855	. . . . . 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 ) )
29	22, 28	impbii 199	. . . . 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 )
30	9, 29	bitri 264	. . . 4 |- ( { a | E. k e. ( 1 ... N ) a = ( A F B ) } C_ RR <-> A. k e. ( 1 ... N ) ( A F B ) e. RR )
31	8, 30	bitri 264	. . 3 |- ( ran ( k e. ( 1 ... N ) |-> ( A F B ) ) C_ RR <-> A. k e. ( 1 ... N ) ( A F B ) e. RR )
32	6, 31	bitri 264	. 2 |- ( ( k e. ( 1 ... N ) |-> ( A F B ) ) : ( 1 ... N ) --> RR <-> A. k e. ( 1 ... N ) ( A F B ) e. RR )
33	1, 32	syl6bb 276	1 |- ( N e. NN -> ( ( k e. ( 1 ... N ) |-> ( A F B ) ) e. ( EE ` N ) <-> A. k e. ( 1 ... N ) ( A F B ) e. RR ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   NNcn 11020   ...cfz 12326   EEcee 25768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ee 25771

This theorem is referenced by:  eleesub  25791  eleesubd  25792  axsegconlem1  25797  axsegconlem8  25804  axpasch  25821  axeuclidlem  25842  axcontlem2  25845