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Theorem recosf1o 19897
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 12405 . . . . . 6  |-  cos : CC
--> CC
2 ffn 5389 . . . . . 6  |-  ( cos
: CC --> CC  ->  cos 
Fn  CC )
31, 2ax-mp 8 . . . . 5  |-  cos  Fn  CC
4 0re 8838 . . . . . . 7  |-  0  e.  RR
5 pire 19832 . . . . . . 7  |-  pi  e.  RR
6 iccssre 10731 . . . . . . 7  |-  ( ( 0  e.  RR  /\  pi  e.  RR )  -> 
( 0 [,] pi )  C_  RR )
74, 5, 6mp2an 653 . . . . . 6  |-  ( 0 [,] pi )  C_  RR
8 ax-resscn 8794 . . . . . 6  |-  RR  C_  CC
97, 8sstri 3188 . . . . 5  |-  ( 0 [,] pi )  C_  CC
10 fnssres 5357 . . . . 5  |-  ( ( cos  Fn  CC  /\  ( 0 [,] pi )  C_  CC )  -> 
( cos  |`  ( 0 [,] pi ) )  Fn  ( 0 [,] pi ) )
113, 9, 10mp2an 653 . . . 4  |-  ( cos  |`  ( 0 [,] pi ) )  Fn  (
0 [,] pi )
12 fvres 5542 . . . . . 6  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( cos `  x
) )
137sseli 3176 . . . . . . 7  |-  ( x  e.  ( 0 [,] pi )  ->  x  e.  RR )
14 cosbnd2 12463 . . . . . . 7  |-  ( x  e.  RR  ->  ( cos `  x )  e.  ( -u 1 [,] 1 ) )
1513, 14syl 15 . . . . . 6  |-  ( x  e.  ( 0 [,] pi )  ->  ( cos `  x )  e.  ( -u 1 [,] 1 ) )
1612, 15eqeltrd 2357 . . . . 5  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  e.  ( -u 1 [,] 1 ) )
1716rgen 2608 . . . 4  |-  A. x  e.  ( 0 [,] pi ) ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  e.  ( -u
1 [,] 1 )
18 ffnfv 5685 . . . 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 886 . . 3  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 )
20 fvres 5542 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 y )  =  ( cos `  y
) )
2112, 20eqeqan12d 2298 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  <-> 
( cos `  x
)  =  ( cos `  y ) ) )
22 cos11 19895 . . . . . 6  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( x  =  y  <->  ( cos `  x
)  =  ( cos `  y ) ) )
2322biimprd 214 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( cos `  x )  =  ( cos `  y )  ->  x  =  y ) )
2421, 23sylbid 206 . . . 4  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  ->  x  =  y ) )
2524rgen2a 2609 . . 3  |-  A. x  e.  ( 0 [,] pi ) A. y  e.  ( 0 [,] pi ) ( ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  ->  x  =  y )
26 dff13 5783 . . 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 886 . 2  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi )
-1-1-> ( -u 1 [,] 1 )
284a1i 10 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  0  e.  RR )
295a1i 10 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  pi  e.  RR )
30 1re 8837 . . . . . . . . 9  |-  1  e.  RR
3130renegcli 9108 . . . . . . . 8  |-  -u 1  e.  RR
3231, 30elicc2i 10716 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  <->  ( x  e.  RR  /\  -u 1  <_  x  /\  x  <_ 
1 ) )
3332simp1bi 970 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  e.  RR )
34 pipos 19833 . . . . . . 7  |-  0  <  pi
3534a1i 10 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  0  <  pi )
369a1i 10 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  (
0 [,] pi ) 
C_  CC )
37 coscn 19821 . . . . . . 7  |-  cos  e.  ( CC -cn-> CC )
3837a1i 10 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  cos  e.  ( CC -cn-> CC ) )
397sseli 3176 . . . . . . . 8  |-  ( z  e.  ( 0 [,] pi )  ->  z  e.  RR )
4039recoscld 12424 . . . . . . 7  |-  ( z  e.  ( 0 [,] pi )  ->  ( cos `  z )  e.  RR )
4140adantl 452 . . . . . 6  |-  ( ( x  e.  ( -u
1 [,] 1 )  /\  z  e.  ( 0 [,] pi ) )  ->  ( cos `  z )  e.  RR )
42 cospi 19840 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
4332simp2bi 971 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  -u 1  <_  x )
4442, 43syl5eqbr 4056 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  ( cos `  pi )  <_  x )
4532simp3bi 972 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  1 )
46 cos0 12430 . . . . . . . 8  |-  ( cos `  0 )  =  1
4745, 46syl6breqr 4063 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  ( cos `  0
) )
4844, 47jca 518 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  (
( cos `  pi )  <_  x  /\  x  <_  ( cos `  0
) ) )
4928, 29, 33, 35, 36, 38, 41, 48ivthle2 18817 . . . . 5  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
50 eqcom 2285 . . . . . . 7  |-  ( x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  =  x )
5120eqeq1d 2291 . . . . . . 7  |-  ( y  e.  ( 0 [,] pi )  ->  (
( ( cos  |`  (
0 [,] pi ) ) `  y )  =  x  <->  ( cos `  y )  =  x ) )
5250, 51syl5bb 248 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
x  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  <->  ( cos `  y )  =  x ) )
5352rexbiia 2576 . . . . 5  |-  ( E. y  e.  ( 0 [,] pi ) x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
5449, 53sylibr 203 . . . 4  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y ) )
5554rgen 2608 . . 3  |-  A. x  e.  ( -u 1 [,] 1 ) E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y )
56 dffo3 5675 . . 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 886 . 2  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi )
-onto-> ( -u 1 [,] 1 )
58 df-f1o 5262 . 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 886 1  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> (
-u 1 [,] 1
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    |` cres 4691    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867    <_ cle 8868   -ucneg 9038   [,]cicc 10659   cosccos 12346   picpi 12348   -cn->ccncf 18380
This theorem is referenced by:  resinf1o  19898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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