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Theorem seff 29733
Description: Let set  S be the real or complex numbers. 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 3995 . 2  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
3 reeff1 13506 . . . . . 6  |-  ( exp  |`  RR ) : RR -1-1-> RR+
4 f1f 5704 . . . . . 6  |-  ( ( exp  |`  RR ) : RR -1-1-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
5 rpssre 11102 . . . . . . 7  |-  RR+  C_  RR
6 fss 5665 . . . . . . 7  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
75, 6mpan2 671 . . . . . 6  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ( exp  |`  RR ) : RR --> RR )
83, 4, 7mp2b 10 . . . . 5  |-  ( exp  |`  RR ) : RR --> RR
9 feq23 5643 . . . . . 6  |-  ( ( S  =  RR  /\  S  =  RR )  ->  ( ( exp  |`  RR ) : S --> S  <->  ( exp  |`  RR ) : RR --> RR ) )
109anidms 645 . . . . 5  |-  ( S  =  RR  ->  (
( exp  |`  RR ) : S --> S  <->  ( exp  |`  RR ) : RR --> RR ) )
118, 10mpbiri 233 . . . 4  |-  ( S  =  RR  ->  ( exp  |`  RR ) : S --> S )
12 reseq2 5203 . . . . 5  |-  ( S  =  RR  ->  ( exp  |`  S )  =  ( exp  |`  RR ) )
1312feq1d 5644 . . . 4  |-  ( S  =  RR  ->  (
( exp  |`  S ) : S --> S  <->  ( exp  |`  RR ) : S --> S ) )
1411, 13mpbird 232 . . 3  |-  ( S  =  RR  ->  ( exp  |`  S ) : S --> S )
15 eff 13469 . . . . . 6  |-  exp : CC
--> CC
16 frel 5660 . . . . . . . . 9  |-  ( exp
: CC --> CC  ->  Rel 
exp )
17 resdm 5246 . . . . . . . . 9  |-  ( Rel 
exp  ->  ( exp  |`  dom  exp )  =  exp )
1815, 16, 17mp2b 10 . . . . . . . 8  |-  ( exp  |`  dom  exp )  =  exp
1915fdmi 5662 . . . . . . . . 9  |-  dom  exp  =  CC
2019reseq2i 5205 . . . . . . . 8  |-  ( exp  |`  dom  exp )  =  ( exp  |`  CC )
2118, 20eqtr3i 2482 . . . . . . 7  |-  exp  =  ( exp  |`  CC )
2221feq1i 5649 . . . . . 6  |-  ( exp
: CC --> CC  <->  ( exp  |`  CC ) : CC --> CC )
2315, 22mpbi 208 . . . . 5  |-  ( exp  |`  CC ) : CC --> CC
24 feq23 5643 . . . . . 6  |-  ( ( S  =  CC  /\  S  =  CC )  ->  ( ( exp  |`  CC ) : S --> S  <->  ( exp  |`  CC ) : CC --> CC ) )
2524anidms 645 . . . . 5  |-  ( S  =  CC  ->  (
( exp  |`  CC ) : S --> S  <->  ( exp  |`  CC ) : CC --> CC ) )
2623, 25mpbiri 233 . . . 4  |-  ( S  =  CC  ->  ( exp  |`  CC ) : S --> S )
27 reseq2 5203 . . . . 5  |-  ( S  =  CC  ->  ( exp  |`  S )  =  ( exp  |`  CC ) )
2827feq1d 5644 . . . 4  |-  ( S  =  CC  ->  (
( exp  |`  S ) : S --> S  <->  ( exp  |`  CC ) : S --> S ) )
2926, 28mpbird 232 . . 3  |-  ( S  =  CC  ->  ( exp  |`  S ) : S --> S )
3014, 29jaoi 379 . 2  |-  ( ( S  =  RR  \/  S  =  CC )  ->  ( exp  |`  S ) : S --> S )
311, 2, 303syl 20 1  |-  ( ph  ->  ( exp  |`  S ) : S --> S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1370    e. wcel 1758    C_ wss 3426   {cpr 3977   dom cdm 4938    |` cres 4940   Rel wrel 4943   -->wf 5512   -1-1->wf1 5513   CCcc 9381   RRcr 9382   RR+crp 11092   expce 13449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-ico 11407  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-fac 12153  df-bc 12180  df-hash 12205  df-shft 12658  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-limsup 13051  df-clim 13068  df-rlim 13069  df-sum 13266  df-ef 13455
This theorem is referenced by: (None)
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