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+