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Theorem eleei 23904
Description: The forward direction of elee 23901. (Contributed by Scott Fenton, 1-Jul-2013.)
Assertion
Ref Expression
eleei  |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ... N ) --> RR )

Proof of Theorem eleei
StepHypRef Expression
1 eleenn 23903 . . 3  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
2 elee 23901 . . 3  |-  ( N  e.  NN  ->  ( A  e.  ( EE `  N )  <->  A :
( 1 ... N
) --> RR ) )
31, 2syl 16 . 2  |-  ( A  e.  ( EE `  N )  ->  ( A  e.  ( EE `  N )  <->  A :
( 1 ... N
) --> RR ) )
43ibi 241 1  |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ... N ) --> RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1767   -->wf 5584   ` cfv 5588  (class class class)co 6284   RRcr 9491   1c1 9493   NNcn 10536   ...cfz 11672   EEcee 23895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-ee 23898
This theorem is referenced by:  eedimeq  23905  fveere  23908  eqeefv  23910
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