Theorem iooref1o 16145

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 9809	. . . . . . . . 9 |- 1 e. RR+
3	2	a1i 9	. . . . . . . 8 |- ( x e. RR -> 1 e. RR+ )
4		rpefcl 12081	. . . . . . . 8 |- ( x e. RR -> ( exp ` x ) e. RR+ )
5	3, 4	rpaddcld 9864	. . . . . . 7 |- ( x e. RR -> ( 1 + ( exp ` x ) ) e. RR+ )
6	5	rpreccld 9859	. . . . . 6 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) e. RR+ )
7	6	rpred 9848	. . . . 5 |- ( x e. RR -> ( 1 / ( 1 + ( exp ` x ) ) ) e. RR )
8	6	rpgt0d 9851	. . . . 5 |- ( x e. RR -> 0 < ( 1 / ( 1 + ( exp ` x ) ) ) )
9		1red 8117	. . . . . . 7 |- ( x e. RR -> 1 e. RR )
10	9, 4	ltaddrpd 9882	. . . . . 6 |- ( x e. RR -> 1 < ( 1 + ( exp ` x ) ) )
11	5	recgt1d 9863	. . . . . 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 8149	. . . . . 6 |- 0 e. RR*
14		1re 8101	. . . . . . 7 |- 1 e. RR
15	14	rexri 8160	. . . . . 6 |- 1 e. RR*
16		elioo2 10073	. . . . . 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 1184	. . . 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 10064	. . . . . . . . . 10 |- ( y e. ( 0 (,) 1 ) -> y e. RR )
21		eliooord 10080	. . . . . . . . . . 11 |- ( y e. ( 0 (,) 1 ) -> ( 0 < y /\ y < 1 ) )
22	21	simpld 112	. . . . . . . . . 10 |- ( y e. ( 0 (,) 1 ) -> 0 < y )
23	20, 22	elrpd 9845	. . . . . . . . 9 |- ( y e. ( 0 (,) 1 ) -> y e. RR+ )
24	23	rpreccld 9859	. . . . . . . 8 |- ( y e. ( 0 (,) 1 ) -> ( 1 / y ) e. RR+ )
25	24	rpred 9848	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> ( 1 / y ) e. RR )
26		1red 8117	. . . . . . 7 |- ( y e. ( 0 (,) 1 ) -> 1 e. RR )
27	25, 26	resubcld 8483	. . . . . 6 |- ( y e. ( 0 (,) 1 ) -> ( ( 1 / y ) - 1 ) e. RR )
28	21	simprd 114	. . . . . . . 8 |- ( y e. ( 0 (,) 1 ) -> y < 1 )
29	23	reclt1d 9862	. . . . . . . 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 8635	. . . . . . 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 9845	. . . . 5 |- ( y e. ( 0 (,) 1 ) -> ( ( 1 / y ) - 1 ) e. RR+ )
34	33	relogcld 15439	. . . 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 8118	. . . . . . 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 9850	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( exp ` x ) e. CC )
39	36, 38	addcld 8122	. . . . . . 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 9850	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> y e. CC )
42	40	rpap0d 9854	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> y # 0 )
43	36, 39, 41, 42	divmulap2d 8927	. . . . . 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 9850	. . . . . . . . 9 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 / y ) e. CC )
46	36, 38, 45	addrsub 8473	. . . . . . . 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 15440	. . . . . . . . 9 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) = ( ( 1 / y ) - 1 ) )
49	48	eqeq2d 2218	. . . . . . . 8 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( ( exp ` x ) = ( exp ` ( log ` ( ( 1 / y ) - 1 ) ) ) <-> ( exp ` x ) = ( ( 1 / y ) - 1 ) ) )
50		reef11 12095	. . . . . . . . 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 2208	. . . . . . 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 9854	. . . . . . 7 |- ( ( x e. RR /\ y e. ( 0 (,) 1 ) ) -> ( 1 + ( exp ` x ) ) # 0 )
57	36, 41, 39, 56	divmulap3d 8928	. . . . . 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 2208	. . . . 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 6169	. 2 |- ( T. -> F : RR -1-1-onto-> ( 0 (,) 1 ) )
63	62	mptru 1382	1 |- F : RR -1-1-onto-> ( 0 (,) 1 )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   T. wtru 1374   e. wcel 2177   class class class wbr 4054   |-> cmpt 4116   -1-1-onto->wf1o 5284   ` cfv 5285  (class class class)co 5962   RRcr 7954   0cc0 7955   1c1 7956   + caddc 7958   x. cmul 7960   RR*cxr 8136   < clt 8137   - cmin 8273   / cdiv 8775   RR+crp 9805   (,)cioo 10040   expce 12038   logclog 15413

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073  ax-arch 8074  ax-caucvg 8075  ax-pre-suploc 8076  ax-addf 8077  ax-mulf 8078

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-disj 4031  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-isom 5294  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-of 6176  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-frec 6495  df-1o 6520  df-oadd 6524  df-er 6638  df-map 6755  df-pm 6756  df-en 6846  df-dom 6847  df-fin 6848  df-sup 7107  df-inf 7108  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-n0 9326  df-z 9403  df-uz 9679  df-q 9771  df-rp 9806  df-xneg 9924  df-xadd 9925  df-ioo 10044  df-ico 10046  df-icc 10047  df-fz 10161  df-fzo 10295  df-seqfrec 10625  df-exp 10716  df-fac 10903  df-bc 10925  df-ihash 10953  df-shft 11211  df-cj 11238  df-re 11239  df-im 11240  df-rsqrt 11394  df-abs 11395  df-clim 11675  df-sumdc 11750  df-ef 12044  df-e 12045  df-rest 13158  df-topgen 13177  df-psmet 14390  df-xmet 14391  df-met 14392  df-bl 14393  df-mopn 14394  df-top 14555  df-topon 14568  df-bases 14600  df-ntr 14653  df-cn 14745  df-cnp 14746  df-tx 14810  df-cncf 15128  df-limced 15213  df-dvap 15214  df-relog 15415

This theorem is referenced by:  iooreen  16146