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Theorem evthicc2 21607
Description: Combine ivthicc 21605 with evthicc 21606 to exactly describe the image of a closed interval. (Contributed by Mario Carneiro, 19-Feb-2015.)
Hypotheses
Ref Expression
evthicc.1  |-  ( ph  ->  A  e.  RR )
evthicc.2  |-  ( ph  ->  B  e.  RR )
evthicc.3  |-  ( ph  ->  A  <_  B )
evthicc.4  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
Assertion
Ref Expression
evthicc2  |-  ( ph  ->  E. x  e.  RR  E. y  e.  RR  ran  F  =  ( x [,] y ) )
Distinct variable groups:    x, y, A    x, B, y    x, F, y    ph, x, y

Proof of Theorem evthicc2
Dummy variables  a 
b  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evthicc.1 . . . 4  |-  ( ph  ->  A  e.  RR )
2 evthicc.2 . . . 4  |-  ( ph  ->  B  e.  RR )
3 evthicc.3 . . . 4  |-  ( ph  ->  A  <_  B )
4 evthicc.4 . . . 4  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
51, 2, 3, 4evthicc 21606 . . 3  |-  ( ph  ->  ( E. a  e.  ( A [,] B
) A. z  e.  ( A [,] B
) ( F `  z )  <_  ( F `  a )  /\  E. b  e.  ( A [,] B ) A. z  e.  ( A [,] B ) ( F `  b
)  <_  ( F `  z ) ) )
6 reeanv 3029 . . 3  |-  ( E. a  e.  ( A [,] B ) E. b  e.  ( A [,] B ) ( A. z  e.  ( A [,] B ) ( F `  z
)  <_  ( F `  a )  /\  A. z  e.  ( A [,] B ) ( F `
 b )  <_ 
( F `  z
) )  <->  ( E. a  e.  ( A [,] B ) A. z  e.  ( A [,] B
) ( F `  z )  <_  ( F `  a )  /\  E. b  e.  ( A [,] B ) A. z  e.  ( A [,] B ) ( F `  b
)  <_  ( F `  z ) ) )
75, 6sylibr 212 . 2  |-  ( ph  ->  E. a  e.  ( A [,] B ) E. b  e.  ( A [,] B ) ( A. z  e.  ( A [,] B
) ( F `  z )  <_  ( F `  a )  /\  A. z  e.  ( A [,] B ) ( F `  b
)  <_  ( F `  z ) ) )
8 r19.26 2989 . . . 4  |-  ( A. z  e.  ( A [,] B ) ( ( F `  z )  <_  ( F `  a )  /\  ( F `  b )  <_  ( F `  z
) )  <->  ( A. z  e.  ( A [,] B ) ( F `
 z )  <_ 
( F `  a
)  /\  A. z  e.  ( A [,] B
) ( F `  b )  <_  ( F `  z )
) )
94adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
10 cncff 21132 . . . . . . . . 9  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
119, 10syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F : ( A [,] B ) --> RR )
12 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  b  e.  ( A [,] B
) )
1311, 12ffvelrnd 6020 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( F `  b )  e.  RR )
1413adantr 465 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  ( F `  b )  e.  RR )
15 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  a  e.  ( A [,] B
) )
1611, 15ffvelrnd 6020 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( F `  a )  e.  RR )
1716adantr 465 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  ( F `  a )  e.  RR )
1811adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  F : ( A [,] B ) --> RR )
19 ffn 5729 . . . . . . . . . 10  |-  ( F : ( A [,] B ) --> RR  ->  F  Fn  ( A [,] B ) )
2018, 19syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  F  Fn  ( A [,] B
) )
2113adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( F `  b )  e.  RR )
2216adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( F `  a )  e.  RR )
23 elicc2 11585 . . . . . . . . . . . . . 14  |-  ( ( ( F `  b
)  e.  RR  /\  ( F `  a )  e.  RR )  -> 
( ( F `  z )  e.  ( ( F `  b
) [,] ( F `
 a ) )  <-> 
( ( F `  z )  e.  RR  /\  ( F `  b
)  <_  ( F `  z )  /\  ( F `  z )  <_  ( F `  a
) ) ) )
2421, 22, 23syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( ( F `  z )  e.  ( ( F `  b ) [,] ( F `  a )
)  <->  ( ( F `
 z )  e.  RR  /\  ( F `
 b )  <_ 
( F `  z
)  /\  ( F `  z )  <_  ( F `  a )
) ) )
25 3anass 977 . . . . . . . . . . . . 13  |-  ( ( ( F `  z
)  e.  RR  /\  ( F `  b )  <_  ( F `  z )  /\  ( F `  z )  <_  ( F `  a
) )  <->  ( ( F `  z )  e.  RR  /\  ( ( F `  b )  <_  ( F `  z )  /\  ( F `  z )  <_  ( F `  a
) ) ) )
2624, 25syl6bb 261 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( ( F `  z )  e.  ( ( F `  b ) [,] ( F `  a )
)  <->  ( ( F `
 z )  e.  RR  /\  ( ( F `  b )  <_  ( F `  z )  /\  ( F `  z )  <_  ( F `  a
) ) ) ) )
27 ancom 450 . . . . . . . . . . . . 13  |-  ( ( ( F `  z
)  <_  ( F `  a )  /\  ( F `  b )  <_  ( F `  z
) )  <->  ( ( F `  b )  <_  ( F `  z
)  /\  ( F `  z )  <_  ( F `  a )
) )
2811ffvelrnda 6019 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  RR )
2928biantrurd 508 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( (
( F `  b
)  <_  ( F `  z )  /\  ( F `  z )  <_  ( F `  a
) )  <->  ( ( F `  z )  e.  RR  /\  ( ( F `  b )  <_  ( F `  z )  /\  ( F `  z )  <_  ( F `  a
) ) ) ) )
3027, 29syl5bb 257 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( (
( F `  z
)  <_  ( F `  a )  /\  ( F `  b )  <_  ( F `  z
) )  <->  ( ( F `  z )  e.  RR  /\  ( ( F `  b )  <_  ( F `  z )  /\  ( F `  z )  <_  ( F `  a
) ) ) ) )
3126, 30bitr4d 256 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( ( F `  z )  e.  ( ( F `  b ) [,] ( F `  a )
)  <->  ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) ) )
3231ralbidva 2900 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( A. z  e.  ( A [,] B ) ( F `  z )  e.  ( ( F `
 b ) [,] ( F `  a
) )  <->  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) ) )
3332biimpar 485 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  A. z  e.  ( A [,] B
) ( F `  z )  e.  ( ( F `  b
) [,] ( F `
 a ) ) )
34 ffnfv 6045 . . . . . . . . 9  |-  ( F : ( A [,] B ) --> ( ( F `  b ) [,] ( F `  a ) )  <->  ( F  Fn  ( A [,] B
)  /\  A. z  e.  ( A [,] B
) ( F `  z )  e.  ( ( F `  b
) [,] ( F `
 a ) ) ) )
3520, 33, 34sylanbrc 664 . . . . . . . 8  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  F : ( A [,] B ) --> ( ( F `  b ) [,] ( F `  a ) ) )
36 frn 5735 . . . . . . . 8  |-  ( F : ( A [,] B ) --> ( ( F `  b ) [,] ( F `  a ) )  ->  ran  F  C_  ( ( F `  b ) [,] ( F `  a
) ) )
3735, 36syl 16 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  ran  F 
C_  ( ( F `
 b ) [,] ( F `  a
) ) )
381adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  A  e.  RR )
392adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  B  e.  RR )
40 ssid 3523 . . . . . . . . . 10  |-  ( A [,] B )  C_  ( A [,] B )
4140a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( A [,] B )  C_  ( A [,] B ) )
42 ax-resscn 9545 . . . . . . . . . . 11  |-  RR  C_  CC
43 ssid 3523 . . . . . . . . . . 11  |-  CC  C_  CC
44 cncfss 21138 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A [,] B
) -cn-> RR )  C_  (
( A [,] B
) -cn-> CC ) )
4542, 43, 44mp2an 672 . . . . . . . . . 10  |-  ( ( A [,] B )
-cn-> RR )  C_  (
( A [,] B
) -cn-> CC )
4645, 9sseldi 3502 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> CC ) )
4711ffvelrnda 6019 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
4838, 39, 12, 15, 41, 46, 47ivthicc 21605 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  (
( F `  b
) [,] ( F `
 a ) ) 
C_  ran  F )
4948adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  (
( F `  b
) [,] ( F `
 a ) ) 
C_  ran  F )
5037, 49eqssd 3521 . . . . . 6  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  ran  F  =  ( ( F `
 b ) [,] ( F `  a
) ) )
51 rspceov 6319 . . . . . 6  |-  ( ( ( F `  b
)  e.  RR  /\  ( F `  a )  e.  RR  /\  ran  F  =  ( ( F `
 b ) [,] ( F `  a
) ) )  ->  E. x  e.  RR  E. y  e.  RR  ran  F  =  ( x [,] y ) )
5214, 17, 50, 51syl3anc 1228 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  A. z  e.  ( A [,] B
) ( ( F `
 z )  <_ 
( F `  a
)  /\  ( F `  b )  <_  ( F `  z )
) )  ->  E. x  e.  RR  E. y  e.  RR  ran  F  =  ( x [,] y
) )
5352ex 434 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( A. z  e.  ( A [,] B ) ( ( F `  z
)  <_  ( F `  a )  /\  ( F `  b )  <_  ( F `  z
) )  ->  E. x  e.  RR  E. y  e.  RR  ran  F  =  ( x [,] y
) ) )
548, 53syl5bir 218 . . 3  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  (
( A. z  e.  ( A [,] B
) ( F `  z )  <_  ( F `  a )  /\  A. z  e.  ( A [,] B ) ( F `  b
)  <_  ( F `  z ) )  ->  E. x  e.  RR  E. y  e.  RR  ran  F  =  ( x [,] y ) ) )
5554rexlimdvva 2962 . 2  |-  ( ph  ->  ( E. a  e.  ( A [,] B
) E. b  e.  ( A [,] B
) ( A. z  e.  ( A [,] B
) ( F `  z )  <_  ( F `  a )  /\  A. z  e.  ( A [,] B ) ( F `  b
)  <_  ( F `  z ) )  ->  E. x  e.  RR  E. y  e.  RR  ran  F  =  ( x [,] y ) ) )
567, 55mpd 15 1  |-  ( ph  ->  E. x  e.  RR  E. y  e.  RR  ran  F  =  ( x [,] y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   class class class wbr 4447   ran crn 5000    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487    <_ cle 9625   [,]cicc 11528   -cn->ccncf 21115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cn 19494  df-cnp 19495  df-cmp 19653  df-tx 19798  df-hmeo 19991  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117
This theorem is referenced by:  dvcnvrelem1  22153
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