Theorem recosf1o 22671

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 13720	. . . . . 6 |- cos : CC --> CC
2		ffn 5730	. . . . . 6 |- ( cos : CC --> CC -> cos Fn CC )
3	1, 2	ax-mp 5	. . . . 5 |- cos Fn CC
4		0re 9595	. . . . . . 7 |- 0 e. RR
5		pire 22601	. . . . . . 7 |- pi e. RR
6		iccssre 11605	. . . . . . 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 9548	. . . . . 6 |- RR C_ CC
9	7, 8	sstri 3513	. . . . 5 |- ( 0 [,] pi ) C_ CC
10		fnssres 5693	. . . . 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 5879	. . . . . 6 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( cos ` x ) )
13	7	sseli 3500	. . . . . . 7 |- ( x e. ( 0 [,] pi ) -> x e. RR )
14		cosbnd2 13778	. . . . . . 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 2555	. . . . 5 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 ) )
17	16	rgen 2824	. . . 4 |- A. x e. ( 0 [,] pi ) ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 )
18		ffnfv 6046	. . . 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 918	. . 3 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 )
20		fvres 5879	. . . . . 6 |- ( y e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` y ) = ( cos ` y ) )
21	12, 20	eqeqan12d 2490	. . . . 5 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( cos ` x ) = ( cos ` y ) ) )
22		cos11 22669	. . . . . 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 2891	. . 3 |- A. x e. ( 0 [,] pi ) A. y e. ( 0 [,] pi ) ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) -> x = y )
26		dff13 6153	. . 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 918	. 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 10639	. . . . . . . 8 |- -u 1 e. RR
31		1re 9594	. . . . . . . 8 |- 1 e. RR
32	30, 31	elicc2i 11589	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) <-> ( x e. RR /\ -u 1 <_ x /\ x <_ 1 ) )
33	32	simp1bi 1011	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> x e. RR )
34		pipos 22603	. . . . . . 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 22590	. . . . . . 7 |- cos e. ( CC -cn-> CC )
38	37	a1i 11	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> cos e. ( CC -cn-> CC ) )
39	7	sseli 3500	. . . . . . . 8 |- ( z e. ( 0 [,] pi ) -> z e. RR )
40	39	recoscld 13739	. . . . . . 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 22614	. . . . . . . 8 |- ( cos ` pi ) = -u 1
43	32	simp2bi 1012	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> -u 1 <_ x )
44	42, 43	syl5eqbr 4480	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) -> ( cos ` pi ) <_ x )
45	32	simp3bi 1013	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> x <_ 1 )
46		cos0 13745	. . . . . . . 8 |- ( cos ` 0 ) = 1
47	45, 46	syl6breqr 4487	. . . . . . 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 21620	. . . . 5 |- ( x e. ( -u 1 [,] 1 ) -> E. y e. ( 0 [,] pi ) ( cos ` y ) = x )
50		eqcom 2476	. . . . . . 7 |- ( x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( ( cos |` ( 0 [,] pi ) ) ` y ) = x )
51	20	eqeq1d 2469	. . . . . . 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 2964	. . . . 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 2824	. . 3 |- A. x e. ( -u 1 [,] 1 ) E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y )
56		dffo3 6035	. . 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 918	. 2 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -onto-> ( -u 1 [,] 1 )
58		df-f1o 5594	. 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 918	1 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> ( -u 1 [,] 1 )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   C_ wss 3476   class class class wbr 4447   |` cres 5001   Fn wfn 5582   -->wf 5583   -1-1->wf1 5584   -onto->wfo 5585   -1-1-onto->wf1o 5586   ` cfv 5587  (class class class)co 6283   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492   < clt 9627   <_ cle 9628   -ucneg 9805   [,]cicc 11531   cosccos 13661   picpi 13663   -cn->ccncf 21131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ioc 11533  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-fac 12321  df-bc 12348  df-hash 12373  df-shft 12862  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-limsup 13256  df-clim 13273  df-rlim 13274  df-sum 13471  df-ef 13664  df-sin 13666  df-cos 13667  df-pi 13669  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-fbas 18203  df-fg 18204  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-lp 19419  df-perf 19420  df-cn 19510  df-cnp 19511  df-haus 19598  df-tx 19814  df-hmeo 20007  df-fil 20098  df-fm 20190  df-flim 20191  df-flf 20192  df-xms 20574  df-ms 20575  df-tms 20576  df-cncf 21133  df-limc 22021  df-dv 22022

This theorem is referenced by:  resinf1o  22672