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Theorem reeff1 12641
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.)
Assertion
Ref Expression
reeff1  |-  ( exp  |`  RR ) : RR -1-1-> RR+

Proof of Theorem reeff1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eff 12604 . . . . 5  |-  exp : CC
--> CC
2 ffn 5524 . . . . 5  |-  ( exp
: CC --> CC  ->  exp 
Fn  CC )
31, 2ax-mp 8 . . . 4  |-  exp  Fn  CC
4 ax-resscn 8973 . . . 4  |-  RR  C_  CC
5 fnssres 5491 . . . 4  |-  ( ( exp  Fn  CC  /\  RR  C_  CC )  -> 
( exp  |`  RR )  Fn  RR )
63, 4, 5mp2an 654 . . 3  |-  ( exp  |`  RR )  Fn  RR
7 fvres 5678 . . . . 5  |-  ( x  e.  RR  ->  (
( exp  |`  RR ) `
 x )  =  ( exp `  x
) )
8 rpefcl 12625 . . . . 5  |-  ( x  e.  RR  ->  ( exp `  x )  e.  RR+ )
97, 8eqeltrd 2454 . . . 4  |-  ( x  e.  RR  ->  (
( exp  |`  RR ) `
 x )  e.  RR+ )
109rgen 2707 . . 3  |-  A. x  e.  RR  ( ( exp  |`  RR ) `  x
)  e.  RR+
11 ffnfv 5826 . . 3  |-  ( ( exp  |`  RR ) : RR --> RR+  <->  ( ( exp  |`  RR )  Fn  RR  /\ 
A. x  e.  RR  ( ( exp  |`  RR ) `
 x )  e.  RR+ ) )
126, 10, 11mpbir2an 887 . 2  |-  ( exp  |`  RR ) : RR --> RR+
13 fvres 5678 . . . . 5  |-  ( y  e.  RR  ->  (
( exp  |`  RR ) `
 y )  =  ( exp `  y
) )
147, 13eqeqan12d 2395 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( exp  |`  RR ) `  x
)  =  ( ( exp  |`  RR ) `  y )  <->  ( exp `  x )  =  ( exp `  y ) ) )
15 reef11 12640 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( exp `  x
)  =  ( exp `  y )  <->  x  =  y ) )
1615biimpd 199 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( exp `  x
)  =  ( exp `  y )  ->  x  =  y ) )
1714, 16sylbid 207 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( ( exp  |`  RR ) `  x
)  =  ( ( exp  |`  RR ) `  y )  ->  x  =  y ) )
1817rgen2a 2708 . 2  |-  A. x  e.  RR  A. y  e.  RR  ( ( ( exp  |`  RR ) `  x )  =  ( ( exp  |`  RR ) `
 y )  ->  x  =  y )
19 dff13 5936 . 2  |-  ( ( 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 )
) )
2012, 18, 19mpbir2an 887 1  |-  ( exp  |`  RR ) : RR -1-1-> RR+
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642    C_ wss 3256    |` cres 4813    Fn wfn 5382   -->wf 5383   -1-1->wf1 5384   ` cfv 5387   CCcc 8914   RRcr 8915   RR+crp 10537   expce 12584
This theorem is referenced by:  reeff1o  20223  seff  27200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-ico 10847  df-fz 10969  df-fzo 11059  df-fl 11122  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-shft 11802  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-limsup 12185  df-clim 12202  df-rlim 12203  df-sum 12400  df-ef 12590
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