Theorem reeff1o 15360

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 12126	. 2 |- ( exp |` RR ) : RR -1-1-> RR+
2		f1f 5503	. . . 4 |- ( ( exp |` RR ) : RR -1-1-> RR+ -> ( exp |` RR ) : RR --> RR+ )
3		ffn 5445	. . . 4 |- ( ( exp |` RR ) : RR --> RR+ -> ( exp |` RR ) Fn RR )
4	1, 2, 3	mp2b 8	. . 3 |- ( exp |` RR ) Fn RR
5		frn 5454	. . . . 5 |- ( ( exp |` RR ) : RR --> RR+ -> ran ( exp |` RR ) C_ RR+ )
6	1, 2, 5	mp2b 8	. . . 4 |- ran ( exp |` RR ) C_ RR+
7		rpre 9817	. . . . . . . . 9 |- ( z e. RR+ -> z e. RR )
8		reeff1olem 15358	. . . . . . . . 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 9822	. . . . . . . . . 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 8154	. . . . . . . . . . 11 |- 0 e. RR*
15		ere 12096	. . . . . . . . . . . 12 |- _e e. RR
16	15	rexri 8165	. . . . . . . . . . 11 |- _e e. RR*
17		elioo2 10078	. . . . . . . . . . 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 15359	. . . . . . . . . 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 1250	. . . . . . . 8 |- ( ( z e. RR+ /\ z < _e ) -> E. x e. RR ( exp ` x ) = z )
22		1lt2 9241	. . . . . . . . . 10 |- 1 < 2
23		egt2lt3 12206	. . . . . . . . . . 11 |- ( 2 < _e /\ _e < 3 )
24	23	simpli 111	. . . . . . . . . 10 |- 2 < _e
25		1re 8106	. . . . . . . . . . 11 |- 1 e. RR
26		2re 9141	. . . . . . . . . . 11 |- 2 e. RR
27	25, 26, 15	lttri 8212	. . . . . . . . . 10 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
28	22, 24, 27	mp2an 426	. . . . . . . . 9 |- 1 < _e
29		1red 8122	. . . . . . . . . 10 |- ( z e. RR+ -> 1 e. RR )
30	15	a1i 9	. . . . . . . . . 10 |- ( z e. RR+ -> _e e. RR )
31		axltwlin 8175	. . . . . . . . . 10 |- ( ( 1 e. RR /\ _e e. RR /\ z e. RR ) -> ( 1 < _e -> ( 1 < z \/ z < _e ) ) )
32	29, 30, 7, 31	syl3anc 1250	. . . . . . . . 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 800	. . . . . . 7 |- ( z e. RR+ -> E. x e. RR ( exp ` x ) = z )
35		fvres 5623	. . . . . . . . 9 |- ( x e. RR -> ( ( exp |` RR ) ` x ) = ( exp ` x ) )
36	35	eqeq1d 2216	. . . . . . . 8 |- ( x e. RR -> ( ( ( exp |` RR ) ` x ) = z <-> ( exp ` x ) = z ) )
37	36	rexbiia 2523	. . . . . . 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 5649	. . . . . . 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 3205	. . . 4 |- RR+ C_ ran ( exp |` RR )
43	6, 42	eqssi 3217	. . 3 |- ran ( exp |` RR ) = RR+
44		df-fo 5296	. . 3 |- ( ( exp |` RR ) : RR -onto-> RR+ <-> ( ( exp |` RR ) Fn RR /\ ran ( exp |` RR ) = RR+ ) )
45	4, 43, 44	mpbir2an 945	. 2 |- ( exp |` RR ) : RR -onto-> RR+
46		df-f1o 5297	. 2 |- ( ( exp |` RR ) : RR -1-1-onto-> RR+ <-> ( ( exp |` RR ) : RR -1-1-> RR+ /\ ( exp |` RR ) : RR -onto-> RR+ ) )
47	1, 45, 46	mpbir2an 945	1 |- ( exp |` RR ) : RR -1-1-onto-> RR+

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2178   E.wrex 2487   C_ wss 3174   class class class wbr 4059   ran crn 4694   |` cres 4695   Fn wfn 5285   -->wf 5286   -1-1->wf1 5287   -onto->wfo 5288   -1-1-onto->wf1o 5289   ` cfv 5290  (class class class)co 5967   RRcr 7959   0cc0 7960   1c1 7961   RR*cxr 8141   < clt 8142   2c2 9122   3c3 9123   RR+crp 9810   (,)cioo 10045   expce 12068   _eceu 12069

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080  ax-pre-suploc 8081  ax-addf 8082  ax-mulf 8083

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-disj 4036  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-oadd 6529  df-er 6643  df-map 6760  df-pm 6761  df-en 6851  df-dom 6852  df-fin 6853  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-xneg 9929  df-xadd 9930  df-ioo 10049  df-ico 10051  df-icc 10052  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-exp 10721  df-fac 10908  df-bc 10930  df-ihash 10958  df-shft 11241  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-sumdc 11780  df-ef 12074  df-e 12075  df-rest 13188  df-topgen 13207  df-psmet 14420  df-xmet 14421  df-met 14422  df-bl 14423  df-mopn 14424  df-top 14585  df-topon 14598  df-bases 14630  df-ntr 14683  df-cn 14775  df-cnp 14776  df-tx 14840  df-cncf 15158  df-limced 15243  df-dvap 15244

This theorem is referenced by:  reefiso  15364  dfrelog  15447  relogf1o  15448  reeflog  15450