Theorem iooref1o 15136

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 9671	. . . . . . . . 9 |- 1 e. RR+
3	2	a1i 9	. . . . . . . 8 |- ( x e. RR -> 1 e. RR+ )
4		rpefcl 11707	. . . . . . . 8 |- ( x e. RR -> ( exp ` x ) e. RR+ )
5	3, 4	rpaddcld 9726	. . . . . . 7 |- ( x e. RR -> ( 1 + ( exp ` x ) ) e. RR+ )
6	5	rpreccld 9721	. . . . . 6 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) e. RR+ )
7	6	rpred 9710	. . . . 5 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) e. RR )
8	6	rpgt0d 9713	. . . . 5 |- ( x e. RR -> 0 < ( 1 / ( 1 + ( exp ` x ) ) ) )
9		1red 7986	. . . . . . 7 |- ( x e. RR -> 1 e. RR )
10	9, 4	ltaddrpd 9744	. . . . . 6 |- ( x e. RR -> 1 < ( 1 + ( exp ` x ) ) )
11	5	recgt1d 9725	. . . . . 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 8018	. . . . . 6 |- 0 e. RR*
14		1re 7970	. . . . . . 7 |- 1 e. RR
15	14	rexri 8029	. . . . . 6 |- 1 e. RR*
16		elioo2 9935	. . . . . 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 1182	. . . 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 9926	. . . . . . . . . 10 |- ( y e. ( 0 (,) 1 ) -> y e. RR )
21		eliooord 9942	. . . . . . . . . . 11 |- ( y e. ( 0 (,) 1 ) -> ( 0 < y /\ y < 1 ) )
22	21	simpld 112	. . . . . . . . . 10 |- ( y e. ( 0 (,) 1 ) -> 0 < y )
23	20, 22	elrpd 9707	. . . . . . . . 9 |- ( y e. ( 0 (,) 1 ) -> y e. RR+ )
24	23	rpreccld 9721	. . . . . . . 8 |- ( y e. ( 0 (,) 1 ) -> ( 1 / y ) e. RR+ )
25	24	rpred 9710	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> ( 1 / y ) e. RR )
26		1red 7986	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> 1 e. RR )
27	25, 26	resubcld 8352	. . . . . 6 |- ( y e. ( 0 (,) 1 ) -> ( ( 1 / y ) - 1 ) e. RR )
28	21	simprd 114	. . . . . . . 8 |- ( y e. ( 0 (,) 1 ) -> y < 1 )
29	23	reclt1d 9724	. . . . . . . 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 8503	. . . . . . 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 9707	. . . . 5 |- ( y e. ( 0 (,) 1 ) -> ( ( 1 / y ) - 1 ) e. RR+ )
34	33	relogcld 14656	. . . 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 7987	. . . . . . 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 9712	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( exp ` x ) e. CC )
39	36, 38	addcld 7991	. . . . . . 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 9712	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> y e. CC )
42	40	rpap0d 9716	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> y # 0 )
43	36, 39, 41, 42	divmulap2d 8795	. . . . . 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 9712	. . . . . . . . 9 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 / y ) e. CC )
46	36, 38, 45	addrsub 8342	. . . . . . . 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 14657	. . . . . . . . 9 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) = ( ( 1 / y ) - 1 ) )
49	48	eqeq2d 2199	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( ( exp ` x ) = ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) <-> ( exp ` x ) = ( ( 1 / y ) - 1 ) ) )
50		reef11 11721	. . . . . . . . 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 2189	. . . . . . 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 9716	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 + ( exp ` x ) ) # 0 )
57	36, 41, 39, 56	divmulap3d 8796	. . . . . 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 2189	. . . . 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 6090	. 2 |- ( T. -> F : RR -1-1-onto-> ( 0 (,) 1 ) )
63	62	mptru 1372	1 |- F : RR -1-1-onto-> ( 0 (,) 1 )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 979   = wceq 1363   T. wtru 1364   e. wcel 2158   class class class wbr 4015   |-> cmpt 4076   -1-1-onto->wf1o 5227   ` cfv 5228  (class class class)co 5888   RRcr 7824   0cc0 7825   1c1 7826   + caddc 7828   x. cmul 7830   RR*cxr 8005   < clt 8006   - cmin 8142   / cdiv 8643   RR+crp 9667   (,)cioo 9902   expce 11664   logclog 14630

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945  ax-pre-suploc 7946  ax-addf 7947  ax-mulf 7948

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-disj 3993  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-of 6097  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-frec 6406  df-1o 6431  df-oadd 6435  df-er 6549  df-map 6664  df-pm 6665  df-en 6755  df-dom 6756  df-fin 6757  df-sup 6997  df-inf 6998  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-xneg 9786  df-xadd 9787  df-ioo 9906  df-ico 9908  df-icc 9909  df-fz 10023  df-fzo 10157  df-seqfrec 10460  df-exp 10534  df-fac 10720  df-bc 10742  df-ihash 10770  df-shft 10838  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-clim 11301  df-sumdc 11376  df-ef 11670  df-e 11671  df-rest 12708  df-topgen 12727  df-psmet 13786  df-xmet 13787  df-met 13788  df-bl 13789  df-mopn 13790  df-top 13851  df-topon 13864  df-bases 13896  df-ntr 13949  df-cn 14041  df-cnp 14042  df-tx 14106  df-cncf 14411  df-limced 14478  df-dvap 14479  df-relog 14632

This theorem is referenced by:  iooreen  15137