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Metamath Proof Explorer

Theorem reeff1

Description: The exponential function maps real arguments one-to-one to positive reals. (Contributed by Steve Rodriguez, 25-Aug-2007) (Revised by Mario Carneiro, 10-Nov-2013)

Ref Expression
Assertion reeff1 
|- ( exp |` RR ) : RR -1-1-> RR+

Proof

Step Hyp Ref Expression
1 eff 
 |-  exp : CC --> CC
2 ffn 
 |-  ( exp : CC --> CC -> exp Fn CC )
3 1 2 ax-mp 
 |-  exp Fn CC
4 ax-resscn 
 |-  RR C_ CC
5 fnssres 
 |-  ( ( exp Fn CC /\ RR C_ CC ) -> ( exp |` RR ) Fn RR )
6 3 4 5 mp2an 
 |-  ( exp |` RR ) Fn RR
7 fvres 
 |-  ( x e. RR -> ( ( exp |` RR ) ` x ) = ( exp ` x ) )
8 rpefcl 
 |-  ( x e. RR -> ( exp ` x ) e. RR+ )
9 7 8 eqeltrd 
 |-  ( x e. RR -> ( ( exp |` RR ) ` x ) e. RR+ )
10 9 rgen 
 |-  A. x e. RR ( ( exp |` RR ) ` x ) e. RR+
11 ffnfv 
 |-  ( ( exp |` RR ) : RR --> RR+ <-> ( ( exp |` RR ) Fn RR /\ A. x e. RR ( ( exp |` RR ) ` x ) e. RR+ ) )
12 6 10 11 mpbir2an 
 |-  ( exp |` RR ) : RR --> RR+
13 fvres 
 |-  ( y e. RR -> ( ( exp |` RR ) ` y ) = ( exp ` y ) )
14 7 13 eqeqan12d 
 |-  ( ( x e. RR /\ y e. RR ) -> ( ( ( exp |` RR ) ` x ) = ( ( exp |` RR ) ` y ) <-> ( exp ` x ) = ( exp ` y ) ) )
15 reef11 
 |-  ( ( x e. RR /\ y e. RR ) -> ( ( exp ` x ) = ( exp ` y ) <-> x = y ) )
16 15 biimpd 
 |-  ( ( x e. RR /\ y e. RR ) -> ( ( exp ` x ) = ( exp ` y ) -> x = y ) )
17 14 16 sylbid 
 |-  ( ( x e. RR /\ y e. RR ) -> ( ( ( exp |` RR ) ` x ) = ( ( exp |` RR ) ` y ) -> x = y ) )
18 17 rgen2 
 |-  A. x e. RR A. y e. RR ( ( ( exp |` RR ) ` x ) = ( ( exp |` RR ) ` y ) -> x = y )
19 dff13 
 |-  ( ( exp |` RR ) : RR -1-1-> RR+ <-> ( ( exp |` RR ) : RR --> RR+ /\ A. x e. RR A. y e. RR ( ( ( exp |` RR ) ` x ) = ( ( exp |` RR ) ` y ) -> x = y ) ) )
20 12 18 19 mpbir2an 
 |-  ( exp |` RR ) : RR -1-1-> RR+