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Theorem recosf1o 21991
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 13409 . . . . . 6  |-  cos : CC
--> CC
2 ffn 5559 . . . . . 6  |-  ( cos
: CC --> CC  ->  cos 
Fn  CC )
31, 2ax-mp 5 . . . . 5  |-  cos  Fn  CC
4 0re 9386 . . . . . . 7  |-  0  e.  RR
5 pire 21921 . . . . . . 7  |-  pi  e.  RR
6 iccssre 11377 . . . . . . 7  |-  ( ( 0  e.  RR  /\  pi  e.  RR )  -> 
( 0 [,] pi )  C_  RR )
74, 5, 6mp2an 672 . . . . . 6  |-  ( 0 [,] pi )  C_  RR
8 ax-resscn 9339 . . . . . 6  |-  RR  C_  CC
97, 8sstri 3365 . . . . 5  |-  ( 0 [,] pi )  C_  CC
10 fnssres 5524 . . . . 5  |-  ( ( cos  Fn  CC  /\  ( 0 [,] pi )  C_  CC )  -> 
( cos  |`  ( 0 [,] pi ) )  Fn  ( 0 [,] pi ) )
113, 9, 10mp2an 672 . . . 4  |-  ( cos  |`  ( 0 [,] pi ) )  Fn  (
0 [,] pi )
12 fvres 5704 . . . . . 6  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( cos `  x
) )
137sseli 3352 . . . . . . 7  |-  ( x  e.  ( 0 [,] pi )  ->  x  e.  RR )
14 cosbnd2 13467 . . . . . . 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 2517 . . . . 5  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  e.  ( -u 1 [,] 1 ) )
1716rgen 2781 . . . 4  |-  A. x  e.  ( 0 [,] pi ) ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  e.  ( -u
1 [,] 1 )
18 ffnfv 5869 . . . 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 911 . . 3  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 )
20 fvres 5704 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 y )  =  ( cos `  y
) )
2112, 20eqeqan12d 2458 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  <-> 
( cos `  x
)  =  ( cos `  y ) ) )
22 cos11 21989 . . . . . 6  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( x  =  y  <->  ( cos `  x
)  =  ( cos `  y ) ) )
2322biimprd 223 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( cos `  x )  =  ( cos `  y )  ->  x  =  y ) )
2421, 23sylbid 215 . . . 4  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  ->  x  =  y ) )
2524rgen2a 2782 . . 3  |-  A. x  e.  ( 0 [,] pi ) A. y  e.  ( 0 [,] pi ) ( ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  ->  x  =  y )
26 dff13 5971 . . 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 911 . 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 neg1rr 10426 . . . . . . . 8  |-  -u 1  e.  RR
31 1re 9385 . . . . . . . 8  |-  1  e.  RR
3230, 31elicc2i 11361 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  <->  ( x  e.  RR  /\  -u 1  <_  x  /\  x  <_ 
1 ) )
3332simp1bi 1003 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  e.  RR )
34 pipos 21923 . . . . . . 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 21910 . . . . . . 7  |-  cos  e.  ( CC -cn-> CC )
3837a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  cos  e.  ( CC -cn-> CC ) )
397sseli 3352 . . . . . . . 8  |-  ( z  e.  ( 0 [,] pi )  ->  z  e.  RR )
4039recoscld 13428 . . . . . . 7  |-  ( z  e.  ( 0 [,] pi )  ->  ( cos `  z )  e.  RR )
4140adantl 466 . . . . . 6  |-  ( ( x  e.  ( -u
1 [,] 1 )  /\  z  e.  ( 0 [,] pi ) )  ->  ( cos `  z )  e.  RR )
42 cospi 21934 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
4332simp2bi 1004 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  -u 1  <_  x )
4442, 43syl5eqbr 4325 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  ( cos `  pi )  <_  x )
4532simp3bi 1005 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  1 )
46 cos0 13434 . . . . . . . 8  |-  ( cos `  0 )  =  1
4745, 46syl6breqr 4332 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  ( cos `  0
) )
4844, 47jca 532 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  (
( cos `  pi )  <_  x  /\  x  <_  ( cos `  0
) ) )
4928, 29, 33, 35, 36, 38, 41, 48ivthle2 20941 . . . . 5  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
50 eqcom 2445 . . . . . . 7  |-  ( x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  =  x )
5120eqeq1d 2451 . . . . . . 7  |-  ( y  e.  ( 0 [,] pi )  ->  (
( ( cos  |`  (
0 [,] pi ) ) `  y )  =  x  <->  ( cos `  y )  =  x ) )
5250, 51syl5bb 257 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
x  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  <->  ( cos `  y )  =  x ) )
5352rexbiia 2748 . . . . 5  |-  ( E. y  e.  ( 0 [,] pi ) x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
5449, 53sylibr 212 . . . 4  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y ) )
5554rgen 2781 . . 3  |-  A. x  e.  ( -u 1 [,] 1 ) E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y )
56 dffo3 5858 . . 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 911 . 2  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi )
-onto-> ( -u 1 [,] 1 )
58 df-f1o 5425 . 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 911 1  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> (
-u 1 [,] 1
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716    C_ wss 3328   class class class wbr 4292    |` cres 4842    Fn wfn 5413   -->wf 5414   -1-1->wf1 5415   -onto->wfo 5416   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    < clt 9418    <_ cle 9419   -ucneg 9596   [,]cicc 11303   cosccos 13350   picpi 13352   -cn->ccncf 20452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342
This theorem is referenced by:  resinf1o  21992
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