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Theorem seff 26950
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 3660 . 2  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
3 reeff1 12400 . . . . . 6  |-  ( exp  |`  RR ) : RR -1-1-> RR+
4 f1f 5437 . . . . . 6  |-  ( ( exp  |`  RR ) : RR -1-1-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
5 rpssre 10364 . . . . . . 7  |-  RR+  C_  RR
6 fss 5397 . . . . . . 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 5378 . . . . . 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 4950 . . . . 5  |-  ( S  =  RR  ->  ( exp  |`  S )  =  ( exp  |`  RR ) )
1312feq1d 5379 . . . 4  |-  ( S  =  RR  ->  (
( exp  |`  S ) : S --> S  <->  ( exp  |`  RR ) : S --> S ) )
1411, 13mpbird 223 . . 3  |-  ( S  =  RR  ->  ( exp  |`  S ) : S --> S )
15 eff 12363 . . . . . 6  |-  exp : CC
--> CC
16 frel 5392 . . . . . . . . 9  |-  ( exp
: CC --> CC  ->  Rel 
exp )
17 resdm 4993 . . . . . . . . 9  |-  ( Rel 
exp  ->  ( exp  |`  dom  exp )  =  exp )
1815, 16, 17mp2b 9 . . . . . . . 8  |-  ( exp  |`  dom  exp )  =  exp
1915fdmi 5394 . . . . . . . . 9  |-  dom  exp  =  CC
2019reseq2i 4952 . . . . . . . 8  |-  ( exp  |`  dom  exp )  =  ( exp  |`  CC )
2118, 20eqtr3i 2305 . . . . . . 7  |-  exp  =  ( exp  |`  CC )
2221feq1i 5383 . . . . . 6  |-  ( exp
: CC --> CC  <->  ( exp  |`  CC ) : CC --> CC )
2315, 22mpbi 199 . . . . 5  |-  ( exp  |`  CC ) : CC --> CC
24 feq23 5378 . . . . . 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 4950 . . . . 5  |-  ( S  =  CC  ->  ( exp  |`  S )  =  ( exp  |`  CC ) )
2827feq1d 5379 . . . 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 1623    e. wcel 1684    C_ wss 3152   {cpr 3641   dom cdm 4689    |` cres 4691   Rel wrel 4694   -->wf 5251   -1-1->wf1 5252   CCcc 8735   RRcr 8736   RR+crp 10354   expce 12343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349
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