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Theorem seff 27641
Description: Let set  S be the reals or complexes. Then the exponential function restricted to  S is a mapping from  S to  S. (Contributed by Steve Rodriguez, 6-Nov-2015.)
Hypothesis
Ref Expression
seff.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
Assertion
Ref Expression
seff  |-  ( ph  ->  ( exp  |`  S ) : S --> S )

Proof of Theorem seff
StepHypRef Expression
1 seff.s . 2  |-  ( ph  ->  S  e.  { RR ,  CC } )
2 elpri 3673 . 2  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
3 reeff1 12416 . . . . . 6  |-  ( exp  |`  RR ) : RR -1-1-> RR+
4 f1f 5453 . . . . . 6  |-  ( ( exp  |`  RR ) : RR -1-1-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
5 rpssre 10380 . . . . . . 7  |-  RR+  C_  RR
6 fss 5413 . . . . . . 7  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
75, 6mpan2 652 . . . . . 6  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ( exp  |`  RR ) : RR --> RR )
83, 4, 7mp2b 9 . . . . 5  |-  ( exp  |`  RR ) : RR --> RR
9 feq23 5394 . . . . . 6  |-  ( ( S  =  RR  /\  S  =  RR )  ->  ( ( exp  |`  RR ) : S --> S  <->  ( exp  |`  RR ) : RR --> RR ) )
109anidms 626 . . . . 5  |-  ( S  =  RR  ->  (
( exp  |`  RR ) : S --> S  <->  ( exp  |`  RR ) : RR --> RR ) )
118, 10mpbiri 224 . . . 4  |-  ( S  =  RR  ->  ( exp  |`  RR ) : S --> S )
12 reseq2 4966 . . . . 5  |-  ( S  =  RR  ->  ( exp  |`  S )  =  ( exp  |`  RR ) )
1312feq1d 5395 . . . 4  |-  ( S  =  RR  ->  (
( exp  |`  S ) : S --> S  <->  ( exp  |`  RR ) : S --> S ) )
1411, 13mpbird 223 . . 3  |-  ( S  =  RR  ->  ( exp  |`  S ) : S --> S )
15 eff 12379 . . . . . 6  |-  exp : CC
--> CC
16 frel 5408 . . . . . . . . 9  |-  ( exp
: CC --> CC  ->  Rel 
exp )
17 resdm 5009 . . . . . . . . 9  |-  ( Rel 
exp  ->  ( exp  |`  dom  exp )  =  exp )
1815, 16, 17mp2b 9 . . . . . . . 8  |-  ( exp  |`  dom  exp )  =  exp
1915fdmi 5410 . . . . . . . . 9  |-  dom  exp  =  CC
2019reseq2i 4968 . . . . . . . 8  |-  ( exp  |`  dom  exp )  =  ( exp  |`  CC )
2118, 20eqtr3i 2318 . . . . . . 7  |-  exp  =  ( exp  |`  CC )
2221feq1i 5399 . . . . . 6  |-  ( exp
: CC --> CC  <->  ( exp  |`  CC ) : CC --> CC )
2315, 22mpbi 199 . . . . 5  |-  ( exp  |`  CC ) : CC --> CC
24 feq23 5394 . . . . . 6  |-  ( ( S  =  CC  /\  S  =  CC )  ->  ( ( exp  |`  CC ) : S --> S  <->  ( exp  |`  CC ) : CC --> CC ) )
2524anidms 626 . . . . 5  |-  ( S  =  CC  ->  (
( exp  |`  CC ) : S --> S  <->  ( exp  |`  CC ) : CC --> CC ) )
2623, 25mpbiri 224 . . . 4  |-  ( S  =  CC  ->  ( exp  |`  CC ) : S --> S )
27 reseq2 4966 . . . . 5  |-  ( S  =  CC  ->  ( exp  |`  S )  =  ( exp  |`  CC ) )
2827feq1d 5395 . . . 4  |-  ( S  =  CC  ->  (
( exp  |`  S ) : S --> S  <->  ( exp  |`  CC ) : S --> S ) )
2926, 28mpbird 223 . . 3  |-  ( S  =  CC  ->  ( exp  |`  S ) : S --> S )
3014, 29jaoi 368 . 2  |-  ( ( S  =  RR  \/  S  =  CC )  ->  ( exp  |`  S ) : S --> S )
311, 2, 303syl 18 1  |-  ( ph  ->  ( exp  |`  S ) : S --> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1632    e. wcel 1696    C_ wss 3165   {cpr 3654   dom cdm 4705    |` cres 4707   Rel wrel 4710   -->wf 5267   -1-1->wf1 5268   CCcc 8751   RRcr 8752   RR+crp 10370   expce 12359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365
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