Theorem iooref1o 14323

Description: A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)

Hypothesis

Ref	Expression
iooref1o.f	|- F = ( x e. RR |-> ( 1 / ( 1 + ( exp ` x ) ) ) )

Assertion

Ref	Expression
iooref1o	|- F : RR -1-1-onto-> ( 0 (,) 1 )

Proof of Theorem iooref1o

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iooref1o.f	. . 3 |- F = ( x e. RR |-> ( 1 / ( 1 + ( exp ` x ) ) ) )
2		1rp 9626	. . . . . . . . 9 |- 1 e. RR+
3	2	a1i 9	. . . . . . . 8 |- ( x e. RR -> 1 e. RR+ )
4		rpefcl 11659	. . . . . . . 8 |- ( x e. RR -> ( exp ` x ) e. RR+ )
5	3, 4	rpaddcld 9681	. . . . . . 7 |- ( x e. RR -> ( 1 + ( exp ` x ) ) e. RR+ )
6	5	rpreccld 9676	. . . . . 6 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) e. RR+ )
7	6	rpred 9665	. . . . 5 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) e. RR )
8	6	rpgt0d 9668	. . . . 5 |- ( x e. RR -> 0 < ( 1 / ( 1 + ( exp ` x ) ) ) )
9		1red 7947	. . . . . . 7 |- ( x e. RR -> 1 e. RR )
10	9, 4	ltaddrpd 9699	. . . . . 6 |- ( x e. RR -> 1 < ( 1 + ( exp ` x ) ) )
11	5	recgt1d 9680	. . . . . 6 |- ( x e. RR -> ( 1 < ( 1 + ( exp ` x ) ) <-> ( 1 / ( 1 + ( exp ` x ) ) ) < 1 ) )
12	10, 11	mpbid 147	. . . . 5 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) < 1 )
13		0xr 7978	. . . . . 6 |- 0 e. RR*
14		1re 7931	. . . . . . 7 |- 1 e. RR
15	14	rexri 7989	. . . . . 6 |- 1 e. RR*
16		elioo2 9890	. . . . . 6 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( ( 1 / ( 1 + ( exp ` x ) ) ) e. ( 0 (,) 1 ) <-> ( ( 1 / ( 1 + ( exp ` x ) ) ) e. RR /\ 0 < ( 1 / ( 1 + ( exp ` x ) ) ) /\ ( 1 / ( 1 + ( exp ` x ) ) ) < 1 ) ) )
17	13, 15, 16	mp2an 426	. . . . 5 |- ( ( 1 / ( 1 + ( exp ` x ) ) ) e. ( 0 (,) 1 ) <-> ( ( 1 / ( 1 + ( exp ` x ) ) ) e. RR /\ 0 < ( 1 / ( 1 + ( exp ` x ) ) ) /\ ( 1 / ( 1 + ( exp ` x ) ) ) < 1 ) )
18	7, 8, 12, 17	syl3anbrc 1181	. . . 4 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) e. ( 0 (,) 1 ) )
19	18	adantl 277	. . 3 |- ( ( T. /\ x e. RR ) -> ( 1 / ( 1 + ( exp ` x ) ) ) e. ( 0 (,) 1 ) )
20		elioore 9881	. . . . . . . . . 10 |- ( y e. ( 0 (,) 1 ) -> y e. RR )
21		eliooord 9897	. . . . . . . . . . 11 |- ( y e. ( 0 (,) 1 ) -> ( 0 < y /\ y < 1 ) )
22	21	simpld 112	. . . . . . . . . 10 |- ( y e. ( 0 (,) 1 ) -> 0 < y )
23	20, 22	elrpd 9662	. . . . . . . . 9 |- ( y e. ( 0 (,) 1 ) -> y e. RR+ )
24	23	rpreccld 9676	. . . . . . . 8 |- ( y e. ( 0 (,) 1 ) -> ( 1 / y ) e. RR+ )
25	24	rpred 9665	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> ( 1 / y ) e. RR )
26		1red 7947	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> 1 e. RR )
27	25, 26	resubcld 8312	. . . . . 6 |- ( y e. ( 0 (,) 1 ) -> ( ( 1 / y ) - 1 ) e. RR )
28	21	simprd 114	. . . . . . . 8 |- ( y e. ( 0 (,) 1 ) -> y < 1 )
29	23	reclt1d 9679	. . . . . . . 8 |- ( y e. ( 0 (,) 1 ) -> ( y < 1 <-> 1 < ( 1 / y ) ) )
30	28, 29	mpbid 147	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> 1 < ( 1 / y ) )
31	26, 25	posdifd 8463	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> ( 1 < ( 1 / y ) <-> 0 < ( ( 1 / y ) - 1 ) ) )
32	30, 31	mpbid 147	. . . . . 6 |- ( y e. ( 0 (,) 1 ) -> 0 < ( ( 1 / y ) - 1 ) )
33	27, 32	elrpd 9662	. . . . 5 |- ( y e. ( 0 (,) 1 ) -> ( ( 1 / y ) - 1 ) e. RR+ )
34	33	relogcld 13854	. . . 4 |- ( y e. ( 0 (,) 1 ) -> ( log ` ( ( 1 / y ) - 1 ) ) e. RR )
35	34	adantl 277	. . 3 |- ( ( T. /\ y e. ( 0 (,) 1 ) ) -> ( log ` ( ( 1 / y ) - 1 ) ) e. RR )
36		1cnd 7948	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> 1 e. CC )
37	4	adantr 276	. . . . . . . . 9 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( exp ` x ) e. RR+ )
38	37	rpcnd 9667	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( exp ` x ) e. CC )
39	36, 38	addcld 7951	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 + ( exp ` x ) ) e. CC )
40	23	adantl 277	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> y e. RR+ )
41	40	rpcnd 9667	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> y e. CC )
42	40	rpap0d 9671	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> y # 0 )
43	36, 39, 41, 42	divmulap2d 8753	. . . . . 6 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( ( 1 / y ) = ( 1 + ( exp ` x ) ) <-> 1 = ( y x. ( 1 + ( exp ` x ) ) ) ) )
44	24	adantl 277	. . . . . . . . . 10 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 / y ) e. RR+ )
45	44	rpcnd 9667	. . . . . . . . 9 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 / y ) e. CC )
46	36, 38, 45	addrsub 8302	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( ( 1 + ( exp ` x ) ) = ( 1 / y ) <-> ( exp ` x ) = ( ( 1 / y ) - 1 ) ) )
47	33	adantl 277	. . . . . . . . . 10 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( ( 1 / y ) - 1 ) e. RR+ )
48	47	reeflogd 13855	. . . . . . . . 9 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) = ( ( 1 / y ) - 1 ) )
49	48	eqeq2d 2187	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( ( exp ` x ) = ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) <-> ( exp ` x ) = ( ( 1 / y ) - 1 ) ) )
50		reef11 11673	. . . . . . . . 9 |- ( ( x e. RR /\ ( log ` ( ( 1 / y ) - 1 ) ) e. RR ) -> ( ( exp ` x ) = ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) <-> x = ( log ` ( ( 1 / y ) - 1 ) ) ) )
51	34, 50	sylan2 286	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( ( exp ` x ) = ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) <-> x = ( log ` ( ( 1 / y ) - 1 ) ) ) )
52	46, 49, 51	3bitr2rd 217	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( x = ( log ` ( ( 1 / y ) - 1 ) ) <-> ( 1 + ( exp ` x ) ) = ( 1 / y ) ) )
53		eqcom 2177	. . . . . . 7 |- ( ( 1 + ( exp ` x ) ) = ( 1 / y ) <-> ( 1 / y ) = ( 1 + ( exp ` x ) ) )
54	52, 53	bitrdi 196	. . . . . 6 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( x = ( log ` ( ( 1 / y ) - 1 ) ) <-> ( 1 / y ) = ( 1 + ( exp ` x ) ) ) )
55	5	adantr 276	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 + ( exp ` x ) ) e. RR+ )
56	55	rpap0d 9671	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 + ( exp ` x ) ) # 0 )
57	36, 41, 39, 56	divmulap3d 8754	. . . . . 6 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( ( 1 / ( 1 + ( exp ` x ) ) ) = y <-> 1 = ( y x. ( 1 + ( exp ` x ) ) ) ) )
58	43, 54, 57	3bitr4d 220	. . . . 5 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( x = ( log ` ( ( 1 / y ) - 1 ) ) <-> ( 1 / ( 1 + ( exp ` x ) ) ) = y ) )
59		eqcom 2177	. . . . 5 |- ( ( 1 / ( 1 + ( exp ` x ) ) ) = y <-> y = ( 1 / ( 1 + ( exp ` x ) ) ) )
60	58, 59	bitrdi 196	. . . 4 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( x = ( log ` ( ( 1 / y ) - 1 ) ) <-> y = ( 1 / ( 1 + ( exp ` x ) ) ) ) )
61	60	adantl 277	. . 3 |- ( ( T. /\ ( x e. RR /\ y e. ( 0 (,) 1 ) ) ) -> ( x = ( log ` ( ( 1 / y ) - 1 ) ) <-> y = ( 1 / ( 1 + ( exp ` x ) ) ) ) )
62	1, 19, 35, 61	f1o2d 6066	. 2 |- ( T. -> F : RR -1-1-onto-> ( 0 (,) 1 ) )
63	62	mptru 1362	1 |- F : RR -1-1-onto-> ( 0 (,) 1 )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   T. wtru 1354   e. wcel 2146   class class class wbr 3998   |-> cmpt 4059   -1-1-onto->wf1o 5207   ` cfv 5208  (class class class)co 5865   RRcr 7785   0cc0 7786   1c1 7787   + caddc 7789   x. cmul 7791   RR*cxr 7965   < clt 7966   - cmin 8102   / cdiv 8601   RR+crp 9622   (,)cioo 9857   expce 11616   logclog 13828

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906  ax-pre-suploc 7907  ax-addf 7908  ax-mulf 7909

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-disj 3976  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-of 6073  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-frec 6382  df-1o 6407  df-oadd 6411  df-er 6525  df-map 6640  df-pm 6641  df-en 6731  df-dom 6732  df-fin 6733  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-xneg 9741  df-xadd 9742  df-ioo 9861  df-ico 9863  df-icc 9864  df-fz 9978  df-fzo 10111  df-seqfrec 10414  df-exp 10488  df-fac 10672  df-bc 10694  df-ihash 10722  df-shft 10790  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-clim 11253  df-sumdc 11328  df-ef 11622  df-e 11623  df-rest 12610  df-topgen 12629  df-psmet 13038  df-xmet 13039  df-met 13040  df-bl 13041  df-mopn 13042  df-top 13047  df-topon 13060  df-bases 13092  df-ntr 13147  df-cn 13239  df-cnp 13240  df-tx 13304  df-cncf 13609  df-limced 13676  df-dvap 13677  df-relog 13830

This theorem is referenced by:  iooreen  14324