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Theorem evthicc2 18820
Description: Combine ivthicc 18818 with evthicc 18819 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 18819 . . 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 2707 . . 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 203 . 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 2675 . . . 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 451 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
10 cncff 18397 . . . . . . . . 9  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
119, 10syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F : ( A [,] B ) --> RR )
12 simprr 733 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  b  e.  ( A [,] B
) )
13 ffvelrn 5663 . . . . . . . 8  |-  ( ( F : ( A [,] B ) --> RR 
/\  b  e.  ( A [,] B ) )  ->  ( F `  b )  e.  RR )
1411, 12, 13syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( F `  b )  e.  RR )
1514adantr 451 . . . . . 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 )
16 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  a  e.  ( A [,] B
) )
17 ffvelrn 5663 . . . . . . . 8  |-  ( ( F : ( A [,] B ) --> RR 
/\  a  e.  ( A [,] B ) )  ->  ( F `  a )  e.  RR )
1811, 16, 17syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( F `  a )  e.  RR )
1918adantr 451 . . . . . 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 )
2011adantr 451 . . . . . . . . . 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 )
21 ffn 5389 . . . . . . . . . 10  |-  ( F : ( A [,] B ) --> RR  ->  F  Fn  ( A [,] B ) )
2220, 21syl 15 . . . . . . . . 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
) )
2314adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( F `  b )  e.  RR )
2418adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( F `  a )  e.  RR )
25 elicc2 10715 . . . . . . . . . . . . . 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
) ) ) )
2623, 24, 25syl2anc 642 . . . . . . . . . . . . 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 )
) ) )
27 3anass 938 . . . . . . . . . . . . 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
) ) ) )
2826, 27syl6bb 252 . . . . . . . . . . . 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
) ) ) ) )
29 ancom 437 . . . . . . . . . . . . 13  |-  ( ( ( F `  z
)  <_  ( F `  a )  /\  ( F `  b )  <_  ( F `  z
) )  <->  ( ( F `  b )  <_  ( F `  z
)  /\  ( F `  z )  <_  ( F `  a )
) )
30 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( F : ( A [,] B ) --> RR 
/\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  RR )
3111, 30sylan 457 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  RR )
3231biantrurd 494 . . . . . . . . . . . . 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
) ) ) ) )
3329, 32syl5bb 248 . . . . . . . . . . . 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
) ) ) ) )
3428, 33bitr4d 247 . . . . . . . . . . 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 )
) ) )
3534ralbidva 2559 . . . . . . . . . 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 )
) ) )
3635biimpar 471 . . . . . . . . 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 ) ) )
37 ffnfv 5685 . . . . . . . . 9  |-  ( F : ( A [,] B ) --> ( ( F `  b ) [,] ( F `  a ) )  <->  ( F  Fn  ( A [,] B
)  /\  A. z  e.  ( A [,] B
) ( F `  z )  e.  ( ( F `  b
) [,] ( F `
 a ) ) ) )
3822, 36, 37sylanbrc 645 . . . . . . . 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 ) ) )
39 frn 5395 . . . . . . . 8  |-  ( F : ( A [,] B ) --> ( ( F `  b ) [,] ( F `  a ) )  ->  ran  F  C_  ( ( F `  b ) [,] ( F `  a
) ) )
4038, 39syl 15 . . . . . . 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
) ) )
411adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  A  e.  RR )
422adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  B  e.  RR )
43 ssid 3197 . . . . . . . . . 10  |-  ( A [,] B )  C_  ( A [,] B )
4443a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  ( A [,] B )  C_  ( A [,] B ) )
45 ax-resscn 8794 . . . . . . . . . . 11  |-  RR  C_  CC
46 ssid 3197 . . . . . . . . . . 11  |-  CC  C_  CC
47 cncfss 18403 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A [,] B
) -cn-> RR )  C_  (
( A [,] B
) -cn-> CC ) )
4845, 46, 47mp2an 653 . . . . . . . . . 10  |-  ( ( A [,] B )
-cn-> RR )  C_  (
( A [,] B
) -cn-> CC )
4948, 9sseldi 3178 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> CC ) )
50 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( F : ( A [,] B ) --> RR 
/\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
5111, 50sylan 457 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  ( A [,] B )  /\  b  e.  ( A [,] B ) ) )  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
5241, 42, 12, 16, 44, 49, 51ivthicc 18818 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  (
( F `  b
) [,] ( F `
 a ) ) 
C_  ran  F )
5352adantr 451 . . . . . . 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 )
5440, 53eqssd 3196 . . . . . 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
) ) )
55 rspceov 5893 . . . . . 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 ) )
5615, 19, 54, 55syl3anc 1182 . . . . 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
) )
5756ex 423 . . . 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
) ) )
588, 57syl5bir 209 . . 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 ) ) )
5958rexlimdvva 2674 . 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 ) ) )
607, 59mpd 14 1  |-  ( ph  ->  E. x  e.  RR  E. y  e.  RR  ran  F  =  ( x [,] y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    <_ cle 8868   [,]cicc 10659   -cn->ccncf 18380
This theorem is referenced by:  dvcnvrelem1  19364
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-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-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-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  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-cn 16957  df-cnp 16958  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382
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