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Theorem evthicc2 21075
Description: Combine ivthicc 21073 with evthicc 21074 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 21074 . . 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 2992 . . 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 2953 . . . 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 20600 . . . . . . . . 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 5952 . . . . . . 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 5952 . . . . . . 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 5666 . . . . . . . . . 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 11470 . . . . . . . . . . . . . 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 969 . . . . . . . . . . . . 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 5951 . . . . . . . . . . . . . 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 2843 . . . . . . . . . 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 5977 . . . . . . . . 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 5672 . . . . . . . 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 3482 . . . . . . . . . 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 9449 . . . . . . . . . . 11  |-  RR  C_  CC
43 ssid 3482 . . . . . . . . . . 11  |-  CC  C_  CC
44 cncfss 20606 . . . . . . . . . . 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 3461 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> CC ) )
4711ffvelrnda 5951 . . . . . . . . 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 21073 . . . . . . . 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 3480 . . . . . 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 6236 . . . . . 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 1219 . . . . 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 2952 . 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 965    = wceq 1370    e. wcel 1758   A.wral 2798   E.wrex 2799    C_ wss 3435   class class class wbr 4399   ran crn 4948    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6199   CCcc 9390   RRcr 9391    <_ cle 9529   [,]cicc 11413   -cn->ccncf 20583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-icc 11417  df-fz 11554  df-fzo 11665  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cn 18962  df-cnp 18963  df-cmp 19121  df-tx 19266  df-hmeo 19459  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585
This theorem is referenced by:  dvcnvrelem1  21621
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