Theorem reeff1o 15019

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 11867	. 2 |- ( exp |` RR ) : RR -1-1-> RR+
2		f1f 5464	. . . 4 |- ( ( exp |` RR ) : RR -1-1-> RR+ -> ( exp |` RR ) : RR --> RR+ )
3		ffn 5408	. . . 4 |- ( ( exp |` RR ) : RR --> RR+ -> ( exp |` RR ) Fn RR )
4	1, 2, 3	mp2b 8	. . 3 |- ( exp |` RR ) Fn RR
5		frn 5417	. . . . 5 |- ( ( exp |` RR ) : RR --> RR+ -> ran ( exp |` RR ) C_ RR+ )
6	1, 2, 5	mp2b 8	. . . 4 |- ran ( exp |` RR ) C_ RR+
7		rpre 9737	. . . . . . . . 9 |- ( z e. RR+ -> z e. RR )
8		reeff1olem 15017	. . . . . . . . 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 9742	. . . . . . . . . 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 8075	. . . . . . . . . . 11 |- 0 e. RR*
15		ere 11837	. . . . . . . . . . . 12 |- _e e. RR
16	15	rexri 8086	. . . . . . . . . . 11 |- _e e. RR*
17		elioo2 9998	. . . . . . . . . . 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 15018	. . . . . . . . . 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 9162	. . . . . . . . . 10 |- 1 < 2
23		egt2lt3 11947	. . . . . . . . . . 11 |- ( 2 < _e /\ _e < 3 )
24	23	simpli 111	. . . . . . . . . 10 |- 2 < _e
25		1re 8027	. . . . . . . . . . 11 |- 1 e. RR
26		2re 9062	. . . . . . . . . . 11 |- 2 e. RR
27	25, 26, 15	lttri 8133	. . . . . . . . . 10 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
28	22, 24, 27	mp2an 426	. . . . . . . . 9 |- 1 < _e
29		1red 8043	. . . . . . . . . 10 |- ( z e. RR+ -> 1 e. RR )
30	15	a1i 9	. . . . . . . . . 10 |- ( z e. RR+ -> _e e. RR )
31		axltwlin 8096	. . . . . . . . . 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 5583	. . . . . . . . 9 |- ( x e. RR -> ( ( exp |` RR ) ` x ) = ( exp ` x ) )
36	35	eqeq1d 2205	. . . . . . . 8 |- ( x e. RR -> ( ( ( exp |` RR ) ` x ) = z <-> ( exp ` x ) = z ) )
37	36	rexbiia 2512	. . . . . . 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 5609	. . . . . . 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 3188	. . . 4 |- RR+ C_ ran ( exp |` RR )
43	6, 42	eqssi 3200	. . 3 |- ran ( exp |` RR ) = RR+
44		df-fo 5265	. . 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 5266	. 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 2167   E.wrex 2476   C_ wss 3157   class class class wbr 4034   ran crn 4665   |` cres 4666   Fn wfn 5254   -->wf 5255   -1-1->wf1 5256   -onto->wfo 5257   -1-1-onto->wf1o 5258   ` cfv 5259  (class class class)co 5923   RRcr 7880   0cc0 7881   1c1 7882   RR*cxr 8062   < clt 8063   2c2 9043   3c3 9044   RR+crp 9730   (,)cioo 9965   expce 11809   _eceu 11810

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999  ax-arch 8000  ax-caucvg 8001  ax-pre-suploc 8002  ax-addf 8003  ax-mulf 8004

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-disj 4012  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-of 6136  df-1st 6199  df-2nd 6200  df-recs 6364  df-irdg 6429  df-frec 6450  df-1o 6475  df-oadd 6479  df-er 6593  df-map 6710  df-pm 6711  df-en 6801  df-dom 6802  df-fin 6803  df-sup 7051  df-inf 7052  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-n0 9252  df-z 9329  df-uz 9604  df-q 9696  df-rp 9731  df-xneg 9849  df-xadd 9850  df-ioo 9969  df-ico 9971  df-icc 9972  df-fz 10086  df-fzo 10220  df-seqfrec 10542  df-exp 10633  df-fac 10820  df-bc 10842  df-ihash 10870  df-shft 10982  df-cj 11009  df-re 11010  df-im 11011  df-rsqrt 11165  df-abs 11166  df-clim 11446  df-sumdc 11521  df-ef 11815  df-e 11816  df-rest 12922  df-topgen 12941  df-psmet 14109  df-xmet 14110  df-met 14111  df-bl 14112  df-mopn 14113  df-top 14244  df-topon 14257  df-bases 14289  df-ntr 14342  df-cn 14434  df-cnp 14435  df-tx 14499  df-cncf 14817  df-limced 14902  df-dvap 14903

This theorem is referenced by:  reefiso  15023  dfrelog  15106  relogf1o  15107  reeflog  15109