Theorem reeff1o 15463

Description: The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)

Assertion

Ref	Expression
reeff1o	|- ( exp |` RR ) : RR -1-1-onto-> RR+

Proof of Theorem reeff1o

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reeff1 12227	. 2 |- ( exp |` RR ) : RR -1-1-> RR+
2		f1f 5533	. . . 4 |- ( ( exp |` RR ) : RR -1-1-> RR+ -> ( exp |` RR ) : RR --> RR+ )
3		ffn 5473	. . . 4 |- ( ( exp |` RR ) : RR --> RR+ -> ( exp |` RR ) Fn RR )
4	1, 2, 3	mp2b 8	. . 3 |- ( exp |` RR ) Fn RR
5		frn 5482	. . . . 5 |- ( ( exp |` RR ) : RR --> RR+ -> ran ( exp |` RR ) C_ RR+ )
6	1, 2, 5	mp2b 8	. . . 4 |- ran ( exp |` RR ) C_ RR+
7		rpre 9868	. . . . . . . . 9 |- ( z e. RR+ -> z e. RR )
8		reeff1olem 15461	. . . . . . . . 9 |- ( ( z e. RR /\ 1 < z ) -> E. x e. RR ( exp ` x ) = z )
9	7, 8	sylan 283	. . . . . . . 8 |- ( ( z e. RR+ /\ 1 < z ) -> E. x e. RR ( exp ` x ) = z )
10	7	adantr 276	. . . . . . . . 9 |- ( ( z e. RR+ /\ z < _e ) -> z e. RR )
11		rpgt0 9873	. . . . . . . . . 10 |- ( z e. RR+ -> 0 < z )
12	11	adantr 276	. . . . . . . . 9 |- ( ( z e. RR+ /\ z < _e ) -> 0 < z )
13		simpr 110	. . . . . . . . 9 |- ( ( z e. RR+ /\ z < _e ) -> z < _e )
14		0xr 8204	. . . . . . . . . . 11 |- 0 e. RR*
15		ere 12197	. . . . . . . . . . . 12 |- _e e. RR
16	15	rexri 8215	. . . . . . . . . . 11 |- _e e. RR*
17		elioo2 10129	. . . . . . . . . . 11 |- ( ( 0 e. RR* /\ _e e. RR* ) -> ( z e. ( 0 (,) _e ) <-> ( z e. RR /\ 0 < z /\ z < _e ) ) )
18	14, 16, 17	mp2an 426	. . . . . . . . . 10 |- ( z e. ( 0 (,) _e ) <-> ( z e. RR /\ 0 < z /\ z < _e ) )
19		reeff1oleme 15462	. . . . . . . . . 10 |- ( z e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = z )
20	18, 19	sylbir 135	. . . . . . . . 9 |- ( ( z e. RR /\ 0 < z /\ z < _e ) -> E. x e. RR ( exp ` x ) = z )
21	10, 12, 13, 20	syl3anc 1271	. . . . . . . 8 |- ( ( z e. RR+ /\ z < _e ) -> E. x e. RR ( exp ` x ) = z )
22		1lt2 9291	. . . . . . . . . 10 |- 1 < 2
23		egt2lt3 12307	. . . . . . . . . . 11 |- ( 2 < _e /\ _e < 3 )
24	23	simpli 111	. . . . . . . . . 10 |- 2 < _e
25		1re 8156	. . . . . . . . . . 11 |- 1 e. RR
26		2re 9191	. . . . . . . . . . 11 |- 2 e. RR
27	25, 26, 15	lttri 8262	. . . . . . . . . 10 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
28	22, 24, 27	mp2an 426	. . . . . . . . 9 |- 1 < _e
29		1red 8172	. . . . . . . . . 10 |- ( z e. RR+ -> 1 e. RR )
30	15	a1i 9	. . . . . . . . . 10 |- ( z e. RR+ -> _e e. RR )
31		axltwlin 8225	. . . . . . . . . 10 |- ( ( 1 e. RR /\ _e e. RR /\ z e. RR ) -> ( 1 < _e -> ( 1 < z \/ z < _e ) ) )
32	29, 30, 7, 31	syl3anc 1271	. . . . . . . . 9 |- ( z e. RR+ -> ( 1 < _e -> ( 1 < z \/ z < _e ) ) )
33	28, 32	mpi 15	. . . . . . . 8 |- ( z e. RR+ -> ( 1 < z \/ z < _e ) )
34	9, 21, 33	mpjaodan 803	. . . . . . 7 |- ( z e. RR+ -> E. x e. RR ( exp ` x ) = z )
35		fvres 5653	. . . . . . . . 9 |- ( x e. RR -> ( ( exp |` RR ) ` x ) = ( exp ` x ) )
36	35	eqeq1d 2238	. . . . . . . 8 |- ( x e. RR -> ( ( ( exp |` RR ) ` x ) = z <-> ( exp ` x ) = z ) )
37	36	rexbiia 2545	. . . . . . 7 |- ( E. x e. RR ( ( exp |` RR ) ` x ) = z <-> E. x e. RR ( exp ` x ) = z )
38	34, 37	sylibr 134	. . . . . 6 |- ( z e. RR+ -> E. x e. RR ( ( exp |` RR ) ` x ) = z )
39		fvelrnb 5683	. . . . . . 7 |- ( ( exp |` RR ) Fn RR -> ( z e. ran ( exp |` RR ) <-> E. x e. RR ( ( exp |` RR ) ` x ) = z ) )
40	4, 39	ax-mp 5	. . . . . 6 |- ( z e. ran ( exp |` RR ) <-> E. x e. RR ( ( exp |` RR ) ` x ) = z )
41	38, 40	sylibr 134	. . . . 5 |- ( z e. RR+ -> z e. ran ( exp |` RR ) )
42	41	ssriv 3228	. . . 4 |- RR+ C_ ran ( exp |` RR )
43	6, 42	eqssi 3240	. . 3 |- ran ( exp |` RR ) = RR+
44		df-fo 5324	. . 3 |- ( ( exp |` RR ) : RR -onto-> RR+ <-> ( ( exp |` RR ) Fn RR /\ ran ( exp |` RR ) = RR+ ) )
45	4, 43, 44	mpbir2an 948	. 2 |- ( exp |` RR ) : RR -onto-> RR+
46		df-f1o 5325	. 2 |- ( ( exp |` RR ) : RR -1-1-onto-> RR+ <-> ( ( exp |` RR ) : RR -1-1-> RR+ /\ ( exp |` RR ) : RR -onto-> RR+ ) )
47	1, 45, 46	mpbir2an 948	1 |- ( exp |` RR ) : RR -1-1-onto-> RR+

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   class class class wbr 4083   ran crn 4720   |` cres 4721   Fn wfn 5313   -->wf 5314   -1-1->wf1 5315   -onto->wfo 5316   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6007   RRcr 8009   0cc0 8010   1c1 8011   RR*cxr 8191   < clt 8192   2c2 9172   3c3 9173   RR+crp 9861   (,)cioo 10096   expce 12169   _eceu 12170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130  ax-pre-suploc 8131  ax-addf 8132  ax-mulf 8133

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-disj 4060  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-of 6224  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-map 6805  df-pm 6806  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-xneg 9980  df-xadd 9981  df-ioo 10100  df-ico 10102  df-icc 10103  df-fz 10217  df-fzo 10351  df-seqfrec 10682  df-exp 10773  df-fac 10960  df-bc 10982  df-ihash 11010  df-shft 11342  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-clim 11806  df-sumdc 11881  df-ef 12175  df-e 12176  df-rest 13290  df-topgen 13309  df-psmet 14523  df-xmet 14524  df-met 14525  df-bl 14526  df-mopn 14527  df-top 14688  df-topon 14701  df-bases 14733  df-ntr 14786  df-cn 14878  df-cnp 14879  df-tx 14943  df-cncf 15261  df-limced 15346  df-dvap 15347

This theorem is referenced by:  reefiso  15467  dfrelog  15550  relogf1o  15551  reeflog  15553