Theorem recosf1o 21950

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 13405	. . . . . 6 |- cos : CC --> CC
2		ffn 5556	. . . . . 6 |- ( cos : CC --> CC -> cos Fn CC )
3	1, 2	ax-mp 5	. . . . 5 |- cos Fn CC
4		0re 9382	. . . . . . 7 |- 0 e. RR
5		pire 21880	. . . . . . 7 |- pi e. RR
6		iccssre 11373	. . . . . . 7 |- ( ( 0 e. RR /\ pi e. RR ) -> ( 0 [,] pi ) C_ RR )
7	4, 5, 6	mp2an 667	. . . . . 6 |- ( 0 [,] pi ) C_ RR
8		ax-resscn 9335	. . . . . 6 |- RR C_ CC
9	7, 8	sstri 3362	. . . . 5 |- ( 0 [,] pi ) C_ CC
10		fnssres 5521	. . . . 5 |- ( ( cos Fn CC /\ ( 0 [,] pi ) C_ CC ) -> ( cos |` ( 0 [,] pi ) ) Fn ( 0 [,] pi ) )
11	3, 9, 10	mp2an 667	. . . 4 |- ( cos |` ( 0 [,] pi ) ) Fn ( 0 [,] pi )
12		fvres 5701	. . . . . 6 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( cos ` x ) )
13	7	sseli 3349	. . . . . . 7 |- ( x e. ( 0 [,] pi ) -> x e. RR )
14		cosbnd2 13463	. . . . . . 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 2515	. . . . 5 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 ) )
17	16	rgen 2779	. . . 4 |- A. x e. ( 0 [,] pi ) ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 )
18		ffnfv 5866	. . . 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 906	. . 3 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 )
20		fvres 5701	. . . . . 6 |- ( y e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` y ) = ( cos ` y ) )
21	12, 20	eqeqan12d 2456	. . . . 5 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( cos ` x ) = ( cos ` y ) ) )
22		cos11 21948	. . . . . 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 2780	. . 3 |- A. x e. ( 0 [,] pi ) A. y e. ( 0 [,] pi ) ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) -> x = y )
26		dff13 5968	. . 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 906	. 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 10422	. . . . . . . 8 |- -u 1 e. RR
31		1re 9381	. . . . . . . 8 |- 1 e. RR
32	30, 31	elicc2i 11357	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) <-> ( x e. RR /\ -u 1 <_ x /\ x <_ 1 ) )
33	32	simp1bi 998	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> x e. RR )
34		pipos 21882	. . . . . . 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 21869	. . . . . . 7 |- cos e. ( CC -cn-> CC )
38	37	a1i 11	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> cos e. ( CC -cn-> CC ) )
39	7	sseli 3349	. . . . . . . 8 |- ( z e. ( 0 [,] pi ) -> z e. RR )
40	39	recoscld 13424	. . . . . . 7 |- ( z e. ( 0 [,] pi ) -> ( cos ` z ) e. RR )
41	40	adantl 463	. . . . . 6 |- ( ( x e. ( -u 1 [,] 1 ) /\ z e. ( 0 [,] pi ) ) -> ( cos ` z ) e. RR )
42		cospi 21893	. . . . . . . 8 |- ( cos ` pi ) = -u 1
43	32	simp2bi 999	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> -u 1 <_ x )
44	42, 43	syl5eqbr 4322	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) -> ( cos ` pi ) <_ x )
45	32	simp3bi 1000	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> x <_ 1 )
46		cos0 13430	. . . . . . . 8 |- ( cos ` 0 ) = 1
47	45, 46	syl6breqr 4329	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) -> x <_ ( cos ` 0 ) )
48	44, 47	jca 529	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> ( ( cos ` pi ) <_ x /\ x <_ ( cos ` 0 ) ) )
49	28, 29, 33, 35, 36, 38, 41, 48	ivthle2 20900	. . . . 5 |- ( x e. ( -u 1 [,] 1 ) -> E. y e. ( 0 [,] pi ) ( cos ` y ) = x )
50		eqcom 2443	. . . . . . 7 |- ( x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( ( cos |` ( 0 [,] pi ) ) ` y ) = x )
51	20	eqeq1d 2449	. . . . . . 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 2746	. . . . 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 2779	. . 3 |- A. x e. ( -u 1 [,] 1 ) E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y )
56		dffo3 5855	. . 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 906	. 2 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -onto-> ( -u 1 [,] 1 )
58		df-f1o 5422	. 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 906	1 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> ( -u 1 [,] 1 )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1364   e. wcel 1761   A.wral 2713   E.wrex 2714   C_ wss 3325   class class class wbr 4289   |` cres 4838   Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   < clt 9414   <_ cle 9415   -ucneg 9592   [,]cicc 11299   cosccos 13346   picpi 13348   -cn->ccncf 20411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301

This theorem is referenced by:  resinf1o  21951