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Theorem evthicc2 18836
Description: Combine ivthicc 18834 with evthicc 18835 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 18835 . . 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 2720 . . 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 2688 . . . 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 18413 . . . . . . . . 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 5679 . . . . . . . 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 5679 . . . . . . . 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 5405 . . . . . . . . . 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 10731 . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . 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 2572 . . . . . . . . . 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 5701 . . . . . . . . 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 5411 . . . . . . . 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 3210 . . . . . . . . . 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 8810 . . . . . . . . . . 11  |-  RR  C_  CC
46 ssid 3210 . . . . . . . . . . 11  |-  CC  C_  CC
47 cncfss 18419 . . . . . . . . . . 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 3191 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  b  e.  ( A [,] B ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> CC ) )
50 ffvelrn 5679 . . . . . . . . . 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 18834 . . . . . . . 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 3209 . . . . . 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 5909 . . . . . 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 2687 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   class class class wbr 4039   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752    <_ cle 8884   [,]cicc 10675   -cn->ccncf 18396
This theorem is referenced by:  dvcnvrelem1  19380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398
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