Theorem reeff1o 15584

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 12341	. 2 |- ( exp |` RR ) : RR -1-1-> RR+
2		f1f 5551	. . . 4 |- ( ( exp |` RR ) : RR -1-1-> RR+ -> ( exp |` RR ) : RR --> RR+ )
3		ffn 5489	. . . 4 |- ( ( exp |` RR ) : RR --> RR+ -> ( exp |` RR ) Fn RR )
4	1, 2, 3	mp2b 8	. . 3 |- ( exp |` RR ) Fn RR
5		frn 5498	. . . . 5 |- ( ( exp |` RR ) : RR --> RR+ -> ran ( exp |` RR ) C_ RR+ )
6	1, 2, 5	mp2b 8	. . . 4 |- ran ( exp |` RR ) C_ RR+
7		rpre 9956	. . . . . . . . 9 |- ( z e. RR+ -> z e. RR )
8		reeff1olem 15582	. . . . . . . . 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 9961	. . . . . . . . . 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 8285	. . . . . . . . . . 11 |- 0 e. RR*
15		ere 12311	. . . . . . . . . . . 12 |- _e e. RR
16	15	rexri 8296	. . . . . . . . . . 11 |- _e e. RR*
17		elioo2 10217	. . . . . . . . . . 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 15583	. . . . . . . . . 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 1274	. . . . . . . 8 |- ( ( z e. RR+ /\ z < _e ) -> E. x e. RR ( exp ` x ) = z )
22		1lt2 9372	. . . . . . . . . 10 |- 1 < 2
23		egt2lt3 12421	. . . . . . . . . . 11 |- ( 2 < _e /\ _e < 3 )
24	23	simpli 111	. . . . . . . . . 10 |- 2 < _e
25		1re 8238	. . . . . . . . . . 11 |- 1 e. RR
26		2re 9272	. . . . . . . . . . 11 |- 2 e. RR
27	25, 26, 15	lttri 8343	. . . . . . . . . 10 |- ( ( 1 < 2 /\ 2 < _e ) -> 1 < _e )
28	22, 24, 27	mp2an 426	. . . . . . . . 9 |- 1 < _e
29		1red 8254	. . . . . . . . . 10 |- ( z e. RR+ -> 1 e. RR )
30	15	a1i 9	. . . . . . . . . 10 |- ( z e. RR+ -> _e e. RR )
31		axltwlin 8306	. . . . . . . . . 10 |- ( ( 1 e. RR /\ _e e. RR /\ z e. RR ) -> ( 1 < _e -> ( 1 < z \/ z < _e ) ) )
32	29, 30, 7, 31	syl3anc 1274	. . . . . . . . 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 806	. . . . . . 7 |- ( z e. RR+ -> E. x e. RR ( exp ` x ) = z )
35		fvres 5672	. . . . . . . . 9 |- ( x e. RR -> ( ( exp |` RR ) ` x ) = ( exp ` x ) )
36	35	eqeq1d 2240	. . . . . . . 8 |- ( x e. RR -> ( ( ( exp |` RR ) ` x ) = z <-> ( exp ` x ) = z ) )
37	36	rexbiia 2548	. . . . . . 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 5702	. . . . . . 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 3232	. . . 4 |- RR+ C_ ran ( exp |` RR )
43	6, 42	eqssi 3244	. . 3 |- ran ( exp |` RR ) = RR+
44		df-fo 5339	. . 3 |- ( ( exp |` RR ) : RR -onto-> RR+ <-> ( ( exp |` RR ) Fn RR /\ ran ( exp |` RR ) = RR+ ) )
45	4, 43, 44	mpbir2an 951	. 2 |- ( exp |` RR ) : RR -onto-> RR+
46		df-f1o 5340	. 2 |- ( ( exp |` RR ) : RR -1-1-onto-> RR+ <-> ( ( exp |` RR ) : RR -1-1-> RR+ /\ ( exp |` RR ) : RR -onto-> RR+ ) )
47	1, 45, 46	mpbir2an 951	1 |- ( exp |` RR ) : RR -1-1-onto-> RR+

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   E.wrex 2512   C_ wss 3201   class class class wbr 4093   ran crn 4732   |` cres 4733   Fn wfn 5328   -->wf 5329   -1-1->wf1 5330   -onto->wfo 5331   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   RRcr 8091   0cc0 8092   1c1 8093   RR*cxr 8272   < clt 8273   2c2 9253   3c3 9254   RR+crp 9949   (,)cioo 10184   expce 12283   _eceu 12284

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212  ax-pre-suploc 8213  ax-addf 8214  ax-mulf 8215

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-disj 4070  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-map 6862  df-pm 6863  df-en 6953  df-dom 6954  df-fin 6955  df-sup 7243  df-inf 7244  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-xneg 10068  df-xadd 10069  df-ioo 10188  df-ico 10190  df-icc 10191  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-fac 11051  df-bc 11073  df-ihash 11101  df-shft 11455  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994  df-ef 12289  df-e 12290  df-rest 13404  df-topgen 13423  df-psmet 14639  df-xmet 14640  df-met 14641  df-bl 14642  df-mopn 14643  df-top 14809  df-topon 14822  df-bases 14854  df-ntr 14907  df-cn 14999  df-cnp 15000  df-tx 15064  df-cncf 15382  df-limced 15467  df-dvap 15468

This theorem is referenced by:  reefiso  15588  dfrelog  15671  relogf1o  15672  reeflog  15674