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Theorem recosf1o 20306
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.
StepHypRef Expression
1 cosf 12655 . . . . . 6  |-  cos : CC
--> CC
2 ffn 5533 . . . . . 6  |-  ( cos
: CC --> CC  ->  cos 
Fn  CC )
31, 2ax-mp 8 . . . . 5  |-  cos  Fn  CC
4 0re 9026 . . . . . . 7  |-  0  e.  RR
5 pire 20241 . . . . . . 7  |-  pi  e.  RR
6 iccssre 10926 . . . . . . 7  |-  ( ( 0  e.  RR  /\  pi  e.  RR )  -> 
( 0 [,] pi )  C_  RR )
74, 5, 6mp2an 654 . . . . . 6  |-  ( 0 [,] pi )  C_  RR
8 ax-resscn 8982 . . . . . 6  |-  RR  C_  CC
97, 8sstri 3302 . . . . 5  |-  ( 0 [,] pi )  C_  CC
10 fnssres 5500 . . . . 5  |-  ( ( cos  Fn  CC  /\  ( 0 [,] pi )  C_  CC )  -> 
( cos  |`  ( 0 [,] pi ) )  Fn  ( 0 [,] pi ) )
113, 9, 10mp2an 654 . . . 4  |-  ( cos  |`  ( 0 [,] pi ) )  Fn  (
0 [,] pi )
12 fvres 5687 . . . . . 6  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( cos `  x
) )
137sseli 3289 . . . . . . 7  |-  ( x  e.  ( 0 [,] pi )  ->  x  e.  RR )
14 cosbnd2 12713 . . . . . . 7  |-  ( x  e.  RR  ->  ( cos `  x )  e.  ( -u 1 [,] 1 ) )
1513, 14syl 16 . . . . . 6  |-  ( x  e.  ( 0 [,] pi )  ->  ( cos `  x )  e.  ( -u 1 [,] 1 ) )
1612, 15eqeltrd 2463 . . . . 5  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  e.  ( -u 1 [,] 1 ) )
1716rgen 2716 . . . 4  |-  A. x  e.  ( 0 [,] pi ) ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  e.  ( -u
1 [,] 1 )
18 ffnfv 5835 . . . 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 ) ) )
1911, 17, 18mpbir2an 887 . . 3  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 )
20 fvres 5687 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 y )  =  ( cos `  y
) )
2112, 20eqeqan12d 2404 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  <-> 
( cos `  x
)  =  ( cos `  y ) ) )
22 cos11 20304 . . . . . 6  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( x  =  y  <->  ( cos `  x
)  =  ( cos `  y ) ) )
2322biimprd 215 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( cos `  x )  =  ( cos `  y )  ->  x  =  y ) )
2421, 23sylbid 207 . . . 4  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  ->  x  =  y ) )
2524rgen2a 2717 . . 3  |-  A. x  e.  ( 0 [,] pi ) A. y  e.  ( 0 [,] pi ) ( ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  ->  x  =  y )
26 dff13 5945 . . 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 )
) )
2719, 25, 26mpbir2an 887 . 2  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi )
-1-1-> ( -u 1 [,] 1 )
284a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  0  e.  RR )
295a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  pi  e.  RR )
30 1re 9025 . . . . . . . . 9  |-  1  e.  RR
3130renegcli 9296 . . . . . . . 8  |-  -u 1  e.  RR
3231, 30elicc2i 10910 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  <->  ( x  e.  RR  /\  -u 1  <_  x  /\  x  <_ 
1 ) )
3332simp1bi 972 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  e.  RR )
34 pipos 20242 . . . . . . 7  |-  0  <  pi
3534a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  0  <  pi )
369a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  (
0 [,] pi ) 
C_  CC )
37 coscn 20230 . . . . . . 7  |-  cos  e.  ( CC -cn-> CC )
3837a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  cos  e.  ( CC -cn-> CC ) )
397sseli 3289 . . . . . . . 8  |-  ( z  e.  ( 0 [,] pi )  ->  z  e.  RR )
4039recoscld 12674 . . . . . . 7  |-  ( z  e.  ( 0 [,] pi )  ->  ( cos `  z )  e.  RR )
4140adantl 453 . . . . . 6  |-  ( ( x  e.  ( -u
1 [,] 1 )  /\  z  e.  ( 0 [,] pi ) )  ->  ( cos `  z )  e.  RR )
42 cospi 20249 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
4332simp2bi 973 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  -u 1  <_  x )
4442, 43syl5eqbr 4188 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  ( cos `  pi )  <_  x )
4532simp3bi 974 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  1 )
46 cos0 12680 . . . . . . . 8  |-  ( cos `  0 )  =  1
4745, 46syl6breqr 4195 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  ( cos `  0
) )
4844, 47jca 519 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  (
( cos `  pi )  <_  x  /\  x  <_  ( cos `  0
) ) )
4928, 29, 33, 35, 36, 38, 41, 48ivthle2 19223 . . . . 5  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
50 eqcom 2391 . . . . . . 7  |-  ( x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  =  x )
5120eqeq1d 2397 . . . . . . 7  |-  ( y  e.  ( 0 [,] pi )  ->  (
( ( cos  |`  (
0 [,] pi ) ) `  y )  =  x  <->  ( cos `  y )  =  x ) )
5250, 51syl5bb 249 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
x  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  <->  ( cos `  y )  =  x ) )
5352rexbiia 2684 . . . . 5  |-  ( E. y  e.  ( 0 [,] pi ) x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
5449, 53sylibr 204 . . . 4  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y ) )
5554rgen 2716 . . 3  |-  A. x  e.  ( -u 1 [,] 1 ) E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y )
56 dffo3 5825 . . 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 ) ) )
5719, 55, 56mpbir2an 887 . 2  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi )
-onto-> ( -u 1 [,] 1 )
58 df-f1o 5403 . 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 ) ) )
5927, 57, 58mpbir2an 887 1  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> (
-u 1 [,] 1
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652    C_ wss 3265   class class class wbr 4155    |` cres 4822    Fn wfn 5391   -->wf 5392   -1-1->wf1 5393   -onto->wfo 5394   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    < clt 9055    <_ cle 9056   -ucneg 9226   [,]cicc 10853   cosccos 12596   picpi 12598   -cn->ccncf 18779
This theorem is referenced by:  resinf1o  20307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004  ax-mulf 9005
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-oi 7414  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-ioo 10854  df-ioc 10855  df-ico 10856  df-icc 10857  df-fz 10978  df-fzo 11068  df-fl 11131  df-seq 11253  df-exp 11312  df-fac 11496  df-bc 11523  df-hash 11548  df-shft 11811  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-limsup 12194  df-clim 12211  df-rlim 12212  df-sum 12409  df-ef 12599  df-sin 12601  df-cos 12602  df-pi 12604  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-hom 13482  df-cco 13483  df-rest 13579  df-topn 13580  df-topgen 13596  df-pt 13597  df-prds 13600  df-xrs 13655  df-0g 13656  df-gsum 13657  df-qtop 13662  df-imas 13663  df-xps 13665  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-submnd 14668  df-mulg 14744  df-cntz 15045  df-cmn 15343  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-fbas 16625  df-fg 16626  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cld 17008  df-ntr 17009  df-cls 17010  df-nei 17087  df-lp 17125  df-perf 17126  df-cn 17215  df-cnp 17216  df-haus 17303  df-tx 17517  df-hmeo 17710  df-fil 17801  df-fm 17893  df-flim 17894  df-flf 17895  df-xms 18261  df-ms 18262  df-tms 18263  df-cncf 18781  df-limc 19622  df-dv 19623
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