Theorem recosf1o 21991

Description: The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)

Assertion

Ref	Expression
recosf1o	|- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> ( -u 1 [,] 1 )

Proof of Theorem recosf1o

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cosf 13409	. . . . . 6 |- cos : CC --> CC
2		ffn 5559	. . . . . 6 |- ( cos : CC --> CC -> cos Fn CC )
3	1, 2	ax-mp 5	. . . . 5 |- cos Fn CC
4		0re 9386	. . . . . . 7 |- 0 e. RR
5		pire 21921	. . . . . . 7 |- pi e. RR
6		iccssre 11377	. . . . . . 7 |- ( ( 0 e. RR /\ pi e. RR ) -> ( 0 [,] pi ) C_ RR )
7	4, 5, 6	mp2an 672	. . . . . 6 |- ( 0 [,] pi ) C_ RR
8		ax-resscn 9339	. . . . . 6 |- RR C_ CC
9	7, 8	sstri 3365	. . . . 5 |- ( 0 [,] pi ) C_ CC
10		fnssres 5524	. . . . 5 |- ( ( cos Fn CC /\ ( 0 [,] pi ) C_ CC ) -> ( cos |` ( 0 [,] pi ) ) Fn ( 0 [,] pi ) )
11	3, 9, 10	mp2an 672	. . . 4 |- ( cos |` ( 0 [,] pi ) ) Fn ( 0 [,] pi )
12		fvres 5704	. . . . . 6 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( cos ` x ) )
13	7	sseli 3352	. . . . . . 7 |- ( x e. ( 0 [,] pi ) -> x e. RR )
14		cosbnd2 13467	. . . . . . 7 |- ( x e. RR -> ( cos ` x ) e. ( -u 1 [,] 1 ) )
15	13, 14	syl 16	. . . . . 6 |- ( x e. ( 0 [,] pi ) -> ( cos ` x ) e. ( -u 1 [,] 1 ) )
16	12, 15	eqeltrd 2517	. . . . 5 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 ) )
17	16	rgen 2781	. . . 4 |- A. x e. ( 0 [,] pi ) ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 )
18		ffnfv 5869	. . . 4 |- ( ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 ) <-> ( ( cos |` ( 0 [,] pi ) ) Fn ( 0 [,] pi ) /\ A. x e. ( 0 [,] pi ) ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 ) ) )
19	11, 17, 18	mpbir2an 911	. . 3 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 )
20		fvres 5704	. . . . . 6 |- ( y e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` y ) = ( cos ` y ) )
21	12, 20	eqeqan12d 2458	. . . . 5 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( cos ` x ) = ( cos ` y ) ) )
22		cos11 21989	. . . . . 6 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( x = y <-> ( cos ` x ) = ( cos ` y ) ) )
23	22	biimprd 223	. . . . 5 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( cos ` x ) = ( cos ` y ) -> x = y ) )
24	21, 23	sylbid 215	. . . 4 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) -> x = y ) )
25	24	rgen2a 2782	. . 3 |- A. x e. ( 0 [,] pi ) A. y e. ( 0 [,] pi ) ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) -> x = y )
26		dff13 5971	. . 3 |- ( ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-> ( -u 1 [,] 1 ) <-> ( ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 ) /\ A. x e. ( 0 [,] pi ) A. y e. ( 0 [,] pi ) ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) -> x = y ) ) )
27	19, 25, 26	mpbir2an 911	. 2 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-> ( -u 1 [,] 1 )
28	4	a1i 11	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> 0 e. RR )
29	5	a1i 11	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> pi e. RR )
30		neg1rr 10426	. . . . . . . 8 |- -u 1 e. RR
31		1re 9385	. . . . . . . 8 |- 1 e. RR
32	30, 31	elicc2i 11361	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) <-> ( x e. RR /\ -u 1 <_ x /\ x <_ 1 ) )
33	32	simp1bi 1003	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> x e. RR )
34		pipos 21923	. . . . . . 7 |- 0 < pi
35	34	a1i 11	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> 0 < pi )
36	9	a1i 11	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> ( 0 [,] pi ) C_ CC )
37		coscn 21910	. . . . . . 7 |- cos e. ( CC -cn-> CC )
38	37	a1i 11	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> cos e. ( CC -cn-> CC ) )
39	7	sseli 3352	. . . . . . . 8 |- ( z e. ( 0 [,] pi ) -> z e. RR )
40	39	recoscld 13428	. . . . . . 7 |- ( z e. ( 0 [,] pi ) -> ( cos ` z ) e. RR )
41	40	adantl 466	. . . . . 6 |- ( ( x e. ( -u 1 [,] 1 ) /\ z e. ( 0 [,] pi ) ) -> ( cos ` z ) e. RR )
42		cospi 21934	. . . . . . . 8 |- ( cos ` pi ) = -u 1
43	32	simp2bi 1004	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> -u 1 <_ x )
44	42, 43	syl5eqbr 4325	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) -> ( cos ` pi ) <_ x )
45	32	simp3bi 1005	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> x <_ 1 )
46		cos0 13434	. . . . . . . 8 |- ( cos ` 0 ) = 1
47	45, 46	syl6breqr 4332	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) -> x <_ ( cos ` 0 ) )
48	44, 47	jca 532	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> ( ( cos ` pi ) <_ x /\ x <_ ( cos ` 0 ) ) )
49	28, 29, 33, 35, 36, 38, 41, 48	ivthle2 20941	. . . . 5 |- ( x e. ( -u 1 [,] 1 ) -> E. y e. ( 0 [,] pi ) ( cos ` y ) = x )
50		eqcom 2445	. . . . . . 7 |- ( x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( ( cos |` ( 0 [,] pi ) ) ` y ) = x )
51	20	eqeq1d 2451	. . . . . . 7 |- ( y e. ( 0 [,] pi ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` y ) = x <-> ( cos ` y ) = x ) )
52	50, 51	syl5bb 257	. . . . . 6 |- ( y e. ( 0 [,] pi ) -> ( x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( cos ` y ) = x ) )
53	52	rexbiia 2748	. . . . 5 |- ( E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> E. y e. ( 0 [,] pi ) ( cos ` y ) = x )
54	49, 53	sylibr 212	. . . 4 |- ( x e. ( -u 1 [,] 1 ) -> E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y ) )
55	54	rgen 2781	. . 3 |- A. x e. ( -u 1 [,] 1 ) E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y )
56		dffo3 5858	. . 3 |- ( ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -onto-> ( -u 1 [,] 1 ) <-> ( ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 ) /\ A. x e. ( -u 1 [,] 1 ) E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y ) ) )
57	19, 55, 56	mpbir2an 911	. 2 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -onto-> ( -u 1 [,] 1 )
58		df-f1o 5425	. 2 |- ( ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> ( -u 1 [,] 1 ) <-> ( ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-> ( -u 1 [,] 1 ) /\ ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -onto-> ( -u 1 [,] 1 ) ) )
59	27, 57, 58	mpbir2an 911	1 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> ( -u 1 [,] 1 )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2715   E.wrex 2716   C_ wss 3328   class class class wbr 4292   |` cres 4842   Fn wfn 5413   -->wf 5414   -1-1->wf1 5415   -onto->wfo 5416   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   < clt 9418   <_ cle 9419   -ucneg 9596   [,]cicc 11303   cosccos 13350   picpi 13352   -cn->ccncf 20452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342

This theorem is referenced by:  resinf1o  21992