Theorem recosf1o 23484

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 14179	. . . . . 6 |- cos : CC --> CC
2		ffn 5728	. . . . . 6 |- ( cos : CC --> CC -> cos Fn CC )
3	1, 2	ax-mp 5	. . . . 5 |- cos Fn CC
4		0re 9643	. . . . . . 7 |- 0 e. RR
5		pire 23413	. . . . . . 7 |- pi e. RR
6		iccssre 11716	. . . . . . 7 |- ( ( 0 e. RR /\ pi e. RR ) -> ( 0 [,] pi ) C_ RR )
7	4, 5, 6	mp2an 678	. . . . . 6 |- ( 0 [,] pi ) C_ RR
8		ax-resscn 9596	. . . . . 6 |- RR C_ CC
9	7, 8	sstri 3441	. . . . 5 |- ( 0 [,] pi ) C_ CC
10		fnssres 5689	. . . . 5 |- ( ( cos Fn CC /\ ( 0 [,] pi ) C_ CC ) -> ( cos |` ( 0 [,] pi ) ) Fn ( 0 [,] pi ) )
11	3, 9, 10	mp2an 678	. . . 4 |- ( cos |` ( 0 [,] pi ) ) Fn ( 0 [,] pi )
12		fvres 5879	. . . . . 6 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( cos ` x ) )
13	7	sseli 3428	. . . . . . 7 |- ( x e. ( 0 [,] pi ) -> x e. RR )
14		cosbnd2 14237	. . . . . . 7 |- ( x e. RR -> ( cos ` x ) e. ( -u 1 [,] 1 ) )
15	13, 14	syl 17	. . . . . 6 |- ( x e. ( 0 [,] pi ) -> ( cos ` x ) e. ( -u 1 [,] 1 ) )
16	12, 15	eqeltrd 2529	. . . . 5 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 ) )
17	16	rgen 2747	. . . 4 |- A. x e. ( 0 [,] pi ) ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 )
18		ffnfv 6049	. . . 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 931	. . 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 2467	. . . . 5 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( cos ` x ) = ( cos ` y ) ) )
22		cos11 23482	. . . . . 6 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( x = y <-> ( cos ` x ) = ( cos ` y ) ) )
23	22	biimprd 227	. . . . 5 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( cos ` x ) = ( cos ` y ) -> x = y ) )
24	21, 23	sylbid 219	. . . 4 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) -> x = y ) )
25	24	rgen2a 2815	. . 3 |- A. x e. ( 0 [,] pi ) A. y e. ( 0 [,] pi ) ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) -> x = y )
26		dff13 6159	. . 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 931	. 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 10714	. . . . . . . 8 |- -u 1 e. RR
31		1re 9642	. . . . . . . 8 |- 1 e. RR
32	30, 31	elicc2i 11700	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) <-> ( x e. RR /\ -u 1 <_ x /\ x <_ 1 ) )
33	32	simp1bi 1023	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> x e. RR )
34		pipos 23415	. . . . . . 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 23400	. . . . . . 7 |- cos e. ( CC -cn-> CC )
38	37	a1i 11	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> cos e. ( CC -cn-> CC ) )
39	7	sseli 3428	. . . . . . . 8 |- ( z e. ( 0 [,] pi ) -> z e. RR )
40	39	recoscld 14198	. . . . . . 7 |- ( z e. ( 0 [,] pi ) -> ( cos ` z ) e. RR )
41	40	adantl 468	. . . . . 6 |- ( ( x e. ( -u 1 [,] 1 ) /\ z e. ( 0 [,] pi ) ) -> ( cos ` z ) e. RR )
42		cospi 23427	. . . . . . . 8 |- ( cos ` pi ) = -u 1
43	32	simp2bi 1024	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> -u 1 <_ x )
44	42, 43	syl5eqbr 4436	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) -> ( cos ` pi ) <_ x )
45	32	simp3bi 1025	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> x <_ 1 )
46		cos0 14204	. . . . . . . 8 |- ( cos ` 0 ) = 1
47	45, 46	syl6breqr 4443	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) -> x <_ ( cos ` 0 ) )
48	44, 47	jca 535	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> ( ( cos ` pi ) <_ x /\ x <_ ( cos ` 0 ) ) )
49	28, 29, 33, 35, 36, 38, 41, 48	ivthle2 22408	. . . . 5 |- ( x e. ( -u 1 [,] 1 ) -> E. y e. ( 0 [,] pi ) ( cos ` y ) = x )
50		eqcom 2458	. . . . . . 7 |- ( x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( ( cos |` ( 0 [,] pi ) ) ` y ) = x )
51	20	eqeq1d 2453	. . . . . . 7 |- ( y e. ( 0 [,] pi ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` y ) = x <-> ( cos ` y ) = x ) )
52	50, 51	syl5bb 261	. . . . . 6 |- ( y e. ( 0 [,] pi ) -> ( x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( cos ` y ) = x ) )
53	52	rexbiia 2888	. . . . 5 |- ( E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> E. y e. ( 0 [,] pi ) ( cos ` y ) = x )
54	49, 53	sylibr 216	. . . 4 |- ( x e. ( -u 1 [,] 1 ) -> E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y ) )
55	54	rgen 2747	. . 3 |- A. x e. ( -u 1 [,] 1 ) E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y )
56		dffo3 6037	. . 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 931	. 2 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -onto-> ( -u 1 [,] 1 )
58		df-f1o 5589	. 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 931	1 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> ( -u 1 [,] 1 )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   E.wrex 2738   C_ wss 3404   class class class wbr 4402   |` cres 4836   Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   < clt 9675   <_ cle 9676   -ucneg 9861   [,]cicc 11638   cosccos 14117   picpi 14119   -cn->ccncf 21908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822

This theorem is referenced by:  resinf1o  23485