Theorem reeff1o 14908

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 11843	. 2 |- ( exp |` RR ) : RR -1-1-> RR+
2		f1f 5459	. . . 4 |- ( ( exp |` RR ) : RR -1-1-> RR+ -> ( exp |` RR ) : RR --> RR+ )
3		ffn 5403	. . . 4 |- ( ( exp |` RR ) : RR --> RR+ -> ( exp |` RR ) Fn RR )
4	1, 2, 3	mp2b 8	. . 3 |- ( exp |` RR ) Fn RR
5		frn 5412	. . . . 5 |- ( ( exp |` RR ) : RR --> RR+ -> ran ( exp |` RR ) C_ RR+ )
6	1, 2, 5	mp2b 8	. . . 4 |- ran ( exp |` RR ) C_ RR+
7		rpre 9726	. . . . . . . . 9 |- ( z e. RR+ -> z e. RR )
8		reeff1olem 14906	. . . . . . . . 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 9731	. . . . . . . . . 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 8066	. . . . . . . . . . 11 |- 0 e. RR*
15		ere 11813	. . . . . . . . . . . 12 |- _e e. RR
16	15	rexri 8077	. . . . . . . . . . 11 |- _e e. RR*
17		elioo2 9987	. . . . . . . . . . 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 14907	. . . . . . . . . 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 1249	. . . . . . . 8 |- ( ( z e. RR+ /\ z < _e ) -> E. x e. RR ( exp ` x ) = z )
22		1lt2 9151	. . . . . . . . . 10 |- 1 < 2
23		egt2lt3 11923	. . . . . . . . . . 11 |- ( 2 < _e /\ _e < 3 )
24	23	simpli 111	. . . . . . . . . 10 |- 2 < _e
25		1re 8018	. . . . . . . . . . 11 |- 1 e. RR
26		2re 9052	. . . . . . . . . . 11 |- 2 e. RR
27	25, 26, 15	lttri 8124	. . . . . . . . . 10 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
28	22, 24, 27	mp2an 426	. . . . . . . . 9 |- 1 < _e
29		1red 8034	. . . . . . . . . 10 |- ( z e. RR+ -> 1 e. RR )
30	15	a1i 9	. . . . . . . . . 10 |- ( z e. RR+ -> _e e. RR )
31		axltwlin 8087	. . . . . . . . . 10 |- ( ( 1 e. RR /\ _e e. RR /\ z e. RR ) -> ( 1 < _e -> ( 1 < z \/ z < _e ) ) )
32	29, 30, 7, 31	syl3anc 1249	. . . . . . . . 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 799	. . . . . . 7 |- ( z e. RR+ -> E. x e. RR ( exp ` x ) = z )
35		fvres 5578	. . . . . . . . 9 |- ( x e. RR -> ( ( exp |` RR ) ` x ) = ( exp ` x ) )
36	35	eqeq1d 2202	. . . . . . . 8 |- ( x e. RR -> ( ( ( exp |` RR ) ` x ) = z <-> ( exp ` x ) = z ) )
37	36	rexbiia 2509	. . . . . . 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 5604	. . . . . . 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 3183	. . . 4 |- RR+ C_ ran ( exp |` RR )
43	6, 42	eqssi 3195	. . 3 |- ran ( exp |` RR ) = RR+
44		df-fo 5260	. . 3 |- ( ( exp |` RR ) : RR -onto-> RR+ <-> ( ( exp |` RR ) Fn RR /\ ran ( exp |` RR ) = RR+ ) )
45	4, 43, 44	mpbir2an 944	. 2 |- ( exp |` RR ) : RR -onto-> RR+
46		df-f1o 5261	. 2 |- ( ( exp |` RR ) : RR -1-1-onto-> RR+ <-> ( ( exp |` RR ) : RR -1-1-> RR+ /\ ( exp |` RR ) : RR -onto-> RR+ ) )
47	1, 45, 46	mpbir2an 944	1 |- ( exp |` RR ) : RR -1-1-onto-> RR+

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   C_ wss 3153   class class class wbr 4029   ran crn 4660   |` cres 4661   Fn wfn 5249   -->wf 5250   -1-1->wf1 5251   -onto->wfo 5252   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   RRcr 7871   0cc0 7872   1c1 7873   RR*cxr 8053   < clt 8054   2c2 9033   3c3 9034   RR+crp 9719   (,)cioo 9954   expce 11785   _eceu 11786

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992  ax-pre-suploc 7993  ax-addf 7994  ax-mulf 7995

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-disj 4007  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-map 6704  df-pm 6705  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-xneg 9838  df-xadd 9839  df-ioo 9958  df-ico 9960  df-icc 9961  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-fac 10797  df-bc 10819  df-ihash 10847  df-shft 10959  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497  df-ef 11791  df-e 11792  df-rest 12852  df-topgen 12871  df-psmet 14039  df-xmet 14040  df-met 14041  df-bl 14042  df-mopn 14043  df-top 14166  df-topon 14179  df-bases 14211  df-ntr 14264  df-cn 14356  df-cnp 14357  df-tx 14421  df-cncf 14726  df-limced 14810  df-dvap 14811

This theorem is referenced by:  reefiso  14912  dfrelog  14995  relogf1o  14996  reeflog  14998