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Theorem seff 31357
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 3964 . 2  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
3 reeff1 13857 . . . . . 6  |-  ( exp  |`  RR ) : RR -1-1-> RR+
4 f1f 5689 . . . . . 6  |-  ( ( exp  |`  RR ) : RR -1-1-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
5 rpssre 11149 . . . . . . 7  |-  RR+  C_  RR
6 fss 5647 . . . . . . 7  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
75, 6mpan2 669 . . . . . 6  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ( exp  |`  RR ) : RR --> RR )
83, 4, 7mp2b 10 . . . . 5  |-  ( exp  |`  RR ) : RR --> RR
9 feq23 5624 . . . . . 6  |-  ( ( S  =  RR  /\  S  =  RR )  ->  ( ( exp  |`  RR ) : S --> S  <->  ( exp  |`  RR ) : RR --> RR ) )
109anidms 643 . . . . 5  |-  ( S  =  RR  ->  (
( exp  |`  RR ) : S --> S  <->  ( exp  |`  RR ) : RR --> RR ) )
118, 10mpbiri 233 . . . 4  |-  ( S  =  RR  ->  ( exp  |`  RR ) : S --> S )
12 reseq2 5181 . . . . 5  |-  ( S  =  RR  ->  ( exp  |`  S )  =  ( exp  |`  RR ) )
1312feq1d 5625 . . . 4  |-  ( S  =  RR  ->  (
( exp  |`  S ) : S --> S  <->  ( exp  |`  RR ) : S --> S ) )
1411, 13mpbird 232 . . 3  |-  ( S  =  RR  ->  ( exp  |`  S ) : S --> S )
15 eff 13819 . . . . . 6  |-  exp : CC
--> CC
16 frel 5642 . . . . . . . . 9  |-  ( exp
: CC --> CC  ->  Rel 
exp )
17 resdm 5227 . . . . . . . . 9  |-  ( Rel 
exp  ->  ( exp  |`  dom  exp )  =  exp )
1815, 16, 17mp2b 10 . . . . . . . 8  |-  ( exp  |`  dom  exp )  =  exp
1915fdmi 5644 . . . . . . . . 9  |-  dom  exp  =  CC
2019reseq2i 5183 . . . . . . . 8  |-  ( exp  |`  dom  exp )  =  ( exp  |`  CC )
2118, 20eqtr3i 2413 . . . . . . 7  |-  exp  =  ( exp  |`  CC )
2221feq1i 5631 . . . . . 6  |-  ( exp
: CC --> CC  <->  ( exp  |`  CC ) : CC --> CC )
2315, 22mpbi 208 . . . . 5  |-  ( exp  |`  CC ) : CC --> CC
24 feq23 5624 . . . . . 6  |-  ( ( S  =  CC  /\  S  =  CC )  ->  ( ( exp  |`  CC ) : S --> S  <->  ( exp  |`  CC ) : CC --> CC ) )
2524anidms 643 . . . . 5  |-  ( S  =  CC  ->  (
( exp  |`  CC ) : S --> S  <->  ( exp  |`  CC ) : CC --> CC ) )
2623, 25mpbiri 233 . . . 4  |-  ( S  =  CC  ->  ( exp  |`  CC ) : S --> S )
27 reseq2 5181 . . . . 5  |-  ( S  =  CC  ->  ( exp  |`  S )  =  ( exp  |`  CC ) )
2827feq1d 5625 . . . 4  |-  ( S  =  CC  ->  (
( exp  |`  S ) : S --> S  <->  ( exp  |`  CC ) : S --> S ) )
2926, 28mpbird 232 . . 3  |-  ( S  =  CC  ->  ( exp  |`  S ) : S --> S )
3014, 29jaoi 377 . 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 366    = wceq 1399    e. wcel 1826    C_ wss 3389   {cpr 3946   dom cdm 4913    |` cres 4915   Rel wrel 4918   -->wf 5492   -1-1->wf1 5493   CCcc 9401   RRcr 9402   RR+crp 11139   expce 13799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-ico 11456  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-fac 12256  df-bc 12283  df-hash 12308  df-shft 12902  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-limsup 13296  df-clim 13313  df-rlim 13314  df-sum 13511  df-ef 13805
This theorem is referenced by: (None)
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