Theorem iooref1o 16944

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 10008	. . . . . . . . 9 |- 1 e. RR+
3	2	a1i 9	. . . . . . . 8 |- ( x e. RR -> 1 e. RR+ )
4		rpefcl 12396	. . . . . . . 8 |- ( x e. RR -> ( exp ` x ) e. RR+ )
5	3, 4	rpaddcld 10063	. . . . . . 7 |- ( x e. RR -> ( 1 + ( exp ` x ) ) e. RR+ )
6	5	rpreccld 10058	. . . . . 6 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) e. RR+ )
7	6	rpred 10047	. . . . 5 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) e. RR )
8	6	rpgt0d 10050	. . . . 5 |- ( x e. RR -> 0 < ( 1 / ( 1 + ( exp ` x ) ) ) )
9		1red 8305	. . . . . . 7 |- ( x e. RR -> 1 e. RR )
10	9, 4	ltaddrpd 10081	. . . . . 6 |- ( x e. RR -> 1 < ( 1 + ( exp ` x ) ) )
11	5	recgt1d 10062	. . . . . 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 8336	. . . . . 6 |- 0 e. RR*
14		1re 8289	. . . . . . 7 |- 1 e. RR
15	14	rexri 8347	. . . . . 6 |- 1 e. RR*
16		elioo2 10273	. . . . . 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 1208	. . . 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 10264	. . . . . . . . . 10 |- ( y e. ( 0 (,) 1 ) -> y e. RR )
21		eliooord 10280	. . . . . . . . . . 11 |- ( y e. ( 0 (,) 1 ) -> ( 0 < y /\ y < 1 ) )
22	21	simpld 112	. . . . . . . . . 10 |- ( y e. ( 0 (,) 1 ) -> 0 < y )
23	20, 22	elrpd 10044	. . . . . . . . 9 |- ( y e. ( 0 (,) 1 ) -> y e. RR+ )
24	23	rpreccld 10058	. . . . . . . 8 |- ( y e. ( 0 (,) 1 ) -> ( 1 / y ) e. RR+ )
25	24	rpred 10047	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> ( 1 / y ) e. RR )
26		1red 8305	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> 1 e. RR )
27	25, 26	resubcld 8671	. . . . . 6 |- ( y e. ( 0 (,) 1 ) -> ( ( 1 / y ) - 1 ) e. RR )
28	21	simprd 114	. . . . . . . 8 |- ( y e. ( 0 (,) 1 ) -> y < 1 )
29	23	reclt1d 10061	. . . . . . . 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 8823	. . . . . . 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 10044	. . . . 5 |- ( y e. ( 0 (,) 1 ) -> ( ( 1 / y ) - 1 ) e. RR+ )
34	33	relogcld 15873	. . . 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 8306	. . . . . . 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 10049	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( exp ` x ) e. CC )
39	36, 38	addcld 8309	. . . . . . 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 10049	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> y e. CC )
42	40	rpap0d 10053	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> y # 0 )
43	36, 39, 41, 42	divmulap2d 9115	. . . . . 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 10049	. . . . . . . . 9 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 / y ) e. CC )
46	36, 38, 45	addrsub 8660	. . . . . . . 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 15874	. . . . . . . . 9 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) = ( ( 1 / y ) - 1 ) )
49	48	eqeq2d 2246	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( ( exp ` x ) = ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) <-> ( exp ` x ) = ( ( 1 / y ) - 1 ) ) )
50		reef11 12410	. . . . . . . . 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 2236	. . . . . . 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 10053	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 + ( exp ` x ) ) # 0 )
57	36, 41, 39, 56	divmulap3d 9116	. . . . . 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 2236	. . . . 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 6268	. 2 |- ( T. -> F : RR -1-1-onto-> ( 0 (,) 1 ) )
63	62	mptru 1407	1 |- F : RR -1-1-onto-> ( 0 (,) 1 )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   T. wtru 1399   e. wcel 2205   class class class wbr 4114   |-> cmpt 4176   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   RR*cxr 8323   < clt 8324   - cmin 8460   / cdiv 8963   RR+crp 10004   (,)cioo 10240   expce 12353   logclog 15847

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263  ax-pre-suploc 8264  ax-addf 8265  ax-mulf 8266

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-disj 4091  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-map 6897  df-pm 6898  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-ioo 10244  df-ico 10246  df-icc 10247  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-bc 11135  df-ihash 11164  df-shft 11525  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064  df-ef 12359  df-e 12360  df-rest 13538  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820  df-mopn 14821  df-top 14989  df-topon 15002  df-bases 15034  df-ntr 15087  df-cn 15179  df-cnp 15180  df-tx 15244  df-cncf 15562  df-limced 15647  df-dvap 15648  df-relog 15849

This theorem is referenced by:  iooreen  16945