Theorem recosf1o 24281

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 14855	. . . . . 6 |- cos : CC --> CC
2		ffn 6045	. . . . . 6 |- ( cos : CC --> CC -> cos Fn CC )
3	1, 2	ax-mp 5	. . . . 5 |- cos Fn CC
4		0re 10040	. . . . . . 7 |- 0 e. RR
5		pire 24210	. . . . . . 7 |- pi e. RR
6		iccssre 12255	. . . . . . 7 |- ( ( 0 e. RR /\ pi e. RR ) -> ( 0 [,] pi ) C_ RR )
7	4, 5, 6	mp2an 708	. . . . . 6 |- ( 0 [,] pi ) C_ RR
8		ax-resscn 9993	. . . . . 6 |- RR C_ CC
9	7, 8	sstri 3612	. . . . 5 |- ( 0 [,] pi ) C_ CC
10		fnssres 6004	. . . . 5 |- ( ( cos Fn CC /\ ( 0 [,] pi ) C_ CC ) -> ( cos |` ( 0 [,] pi ) ) Fn ( 0 [,] pi ) )
11	3, 9, 10	mp2an 708	. . . 4 |- ( cos |` ( 0 [,] pi ) ) Fn ( 0 [,] pi )
12		fvres 6207	. . . . . 6 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( cos ` x ) )
13	7	sseli 3599	. . . . . . 7 |- ( x e. ( 0 [,] pi ) -> x e. RR )
14		cosbnd2 14913	. . . . . . 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 2701	. . . . 5 |- ( x e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 ) )
17	16	rgen 2922	. . . 4 |- A. x e. ( 0 [,] pi ) ( ( cos |` ( 0 [,] pi ) ) ` x ) e. ( -u 1 [,] 1 )
18		ffnfv 6388	. . . 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 955	. . 3 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 )
20		fvres 6207	. . . . . 6 |- ( y e. ( 0 [,] pi ) -> ( ( cos |` ( 0 [,] pi ) ) ` y ) = ( cos ` y ) )
21	12, 20	eqeqan12d 2638	. . . . 5 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( cos ` x ) = ( cos ` y ) ) )
22		cos11 24279	. . . . . 6 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( x = y <-> ( cos ` x ) = ( cos ` y ) ) )
23	22	biimprd 238	. . . . 5 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( cos ` x ) = ( cos ` y ) -> x = y ) )
24	21, 23	sylbid 230	. . . 4 |- ( ( x e. ( 0 [,] pi ) /\ y e. ( 0 [,] pi ) ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) -> x = y ) )
25	24	rgen2a 2977	. . 3 |- A. x e. ( 0 [,] pi ) A. y e. ( 0 [,] pi ) ( ( ( cos |` ( 0 [,] pi ) ) ` x ) = ( ( cos |` ( 0 [,] pi ) ) ` y ) -> x = y )
26		dff13 6512	. . 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 955	. 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 11125	. . . . . . . 8 |- -u 1 e. RR
31		1re 10039	. . . . . . . 8 |- 1 e. RR
32	30, 31	elicc2i 12239	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) <-> ( x e. RR /\ -u 1 <_ x /\ x <_ 1 ) )
33	32	simp1bi 1076	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> x e. RR )
34		pipos 24212	. . . . . . 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 24199	. . . . . . 7 |- cos e. ( CC -cn-> CC )
38	37	a1i 11	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> cos e. ( CC -cn-> CC ) )
39	7	sseli 3599	. . . . . . . 8 |- ( z e. ( 0 [,] pi ) -> z e. RR )
40	39	recoscld 14874	. . . . . . 7 |- ( z e. ( 0 [,] pi ) -> ( cos ` z ) e. RR )
41	40	adantl 482	. . . . . 6 |- ( ( x e. ( -u 1 [,] 1 ) /\ z e. ( 0 [,] pi ) ) -> ( cos ` z ) e. RR )
42		cospi 24224	. . . . . . . 8 |- ( cos ` pi ) = -u 1
43	32	simp2bi 1077	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> -u 1 <_ x )
44	42, 43	syl5eqbr 4688	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) -> ( cos ` pi ) <_ x )
45	32	simp3bi 1078	. . . . . . . 8 |- ( x e. ( -u 1 [,] 1 ) -> x <_ 1 )
46		cos0 14880	. . . . . . . 8 |- ( cos ` 0 ) = 1
47	45, 46	syl6breqr 4695	. . . . . . 7 |- ( x e. ( -u 1 [,] 1 ) -> x <_ ( cos ` 0 ) )
48	44, 47	jca 554	. . . . . 6 |- ( x e. ( -u 1 [,] 1 ) -> ( ( cos ` pi ) <_ x /\ x <_ ( cos ` 0 ) ) )
49	28, 29, 33, 35, 36, 38, 41, 48	ivthle2 23226	. . . . 5 |- ( x e. ( -u 1 [,] 1 ) -> E. y e. ( 0 [,] pi ) ( cos ` y ) = x )
50		eqcom 2629	. . . . . . 7 |- ( x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( ( cos |` ( 0 [,] pi ) ) ` y ) = x )
51	20	eqeq1d 2624	. . . . . . 7 |- ( y e. ( 0 [,] pi ) -> ( ( ( cos |` ( 0 [,] pi ) ) ` y ) = x <-> ( cos ` y ) = x ) )
52	50, 51	syl5bb 272	. . . . . 6 |- ( y e. ( 0 [,] pi ) -> ( x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> ( cos ` y ) = x ) )
53	52	rexbiia 3040	. . . . 5 |- ( E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y ) <-> E. y e. ( 0 [,] pi ) ( cos ` y ) = x )
54	49, 53	sylibr 224	. . . 4 |- ( x e. ( -u 1 [,] 1 ) -> E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y ) )
55	54	rgen 2922	. . 3 |- A. x e. ( -u 1 [,] 1 ) E. y e. ( 0 [,] pi ) x = ( ( cos |` ( 0 [,] pi ) ) ` y )
56		dffo3 6374	. . 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 955	. 2 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -onto-> ( -u 1 [,] 1 )
58		df-f1o 5895	. 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 955	1 |- ( cos |` ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> ( -u 1 [,] 1 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |` cres 5116   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   -ucneg 10267   [,]cicc 12178   cosccos 14795   picpi 14797   -cn->ccncf 22679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  resinf1o  24282