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Theorem evthicc2 22162
Description: Combine ivthicc 22160 with evthicc 22161 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 22161 . . 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 2974 . . 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 2933 . . . 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 463 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
10 cncff 21687 . . . . . . . . 9  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
119, 10syl 17 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F : ( A [,] B ) --> RR )
12 simprr 758 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  b  e.  ( A [,] B
) )
1311, 12ffvelrnd 6009 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( F `  b )  e.  RR )
1413adantr 463 . . . . . 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 756 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  a  e.  ( A [,] B
) )
1611, 15ffvelrnd 6009 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( F `  a )  e.  RR )
1716adantr 463 . . . . . 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 463 . . . . . . . . . 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 5713 . . . . . . . . . 10  |-  ( F : ( A [,] B ) --> RR  ->  F  Fn  ( A [,] B ) )
2018, 19syl 17 . . . . . . . . 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 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( F `  b )  e.  RR )
2216adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( F `  a )  e.  RR )
23 elicc2 11641 . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . 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 978 . . . . . . . . . . . . 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 448 . . . . . . . . . . . . 13  |-  ( ( ( F `  z
)  <_  ( F `  a )  /\  ( F `  b )  <_  ( F `  z
) )  <->  ( ( F `  b )  <_  ( F `  z
)  /\  ( F `  z )  <_  ( F `  a )
) )
2811ffvelrnda 6008 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  RR )
2928biantrurd 506 . . . . . . . . . . . . 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 2839 . . . . . . . . . 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 483 . . . . . . . . 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 6035 . . . . . . . . 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 662 . . . . . . . 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 5719 . . . . . . . 8  |-  ( F : ( A [,] B ) --> ( ( F `  b ) [,] ( F `  a ) )  ->  ran  F  C_  ( ( F `  b ) [,] ( F `  a
) ) )
3735, 36syl 17 . . . . . . 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 463 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  A  e.  RR )
392adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  B  e.  RR )
40 ssid 3460 . . . . . . . . . 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 9578 . . . . . . . . . . 11  |-  RR  C_  CC
43 ssid 3460 . . . . . . . . . . 11  |-  CC  C_  CC
44 cncfss 21693 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A [,] B
) -cn-> RR )  C_  (
( A [,] B
) -cn-> CC ) )
4542, 43, 44mp2an 670 . . . . . . . . . 10  |-  ( ( A [,] B )
-cn-> RR )  C_  (
( A [,] B
) -cn-> CC )
4645, 9sseldi 3439 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> CC ) )
4711ffvelrnda 6008 . . . . . . . . 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 22160 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  (
( F `  b
) [,] ( F `
 a ) ) 
C_  ran  F )
4948adantr 463 . . . . . . 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 3458 . . . . . 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 6316 . . . . . 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 1230 . . . . 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 432 . . . 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 2902 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754    C_ wss 3413   class class class wbr 4394   ran crn 4823    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277   CCcc 9519   RRcr 9520    <_ cle 9658   [,]cicc 11584   -cn->ccncf 21670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-fi 7904  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ioo 11585  df-icc 11588  df-fz 11725  df-fzo 11853  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-rest 15035  df-topn 15036  df-0g 15054  df-gsum 15055  df-topgen 15056  df-pt 15057  df-prds 15060  df-xrs 15114  df-qtop 15119  df-imas 15120  df-xps 15122  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-submnd 16289  df-mulg 16382  df-cntz 16677  df-cmn 17122  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-cnfld 18739  df-top 19689  df-bases 19691  df-topon 19692  df-topsp 19693  df-cn 20019  df-cnp 20020  df-cmp 20178  df-tx 20353  df-hmeo 20546  df-xms 21113  df-ms 21114  df-tms 21115  df-cncf 21672
This theorem is referenced by:  dvcnvrelem1  22708
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