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.

Step	Hyp	Ref	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 ) )
5	1, 2, 3, 4	evthicc 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 ) ) )
7	5, 6	sylibr 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 ) ) )
9	4	adantr 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 )
11	9, 10	syl 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 ) )
13	11, 12	ffvelrnd 5952	. . . . . . 7 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( F ` b ) e. RR )
14	13	adantr 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 ) )
16	11, 15	ffvelrnd 5952	. . . . . . 7 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( F ` a ) e. RR )
17	16	adantr 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 )
18	11	adantr 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 ) )
20	18, 19	syl 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 ) )
21	13	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` b ) e. RR )
22	16	adantr 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 ) ) ) )
24	21, 22, 23	syl2anc 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 ) ) ) )
26	24, 25	syl6bb 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 ) ) )
28	11	ffvelrnda 5951	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` z ) e. RR )
29	28	biantrurd 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 ) ) ) ) )
30	27, 29	syl5bb 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 ) ) ) ) )
31	26, 30	bitr4d 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 ) ) ) )
32	31	ralbidva 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 ) ) ) )
33	32	biimpar 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 ) ) ) )
35	20, 33, 34	sylanbrc 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 ) ) )
37	35, 36	syl 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 ) ) )
38	1	adantr 465	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> A e. RR )
39	2	adantr 465	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> B e. RR )
40		ssid 3482	. . . . . . . . . 10 |- ( A [,] B ) C_ ( A [,] B )
41	40	a1i 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 ) )
45	42, 43, 44	mp2an 672	. . . . . . . . . 10 |- ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC )
46	45, 9	sseldi 3461	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> F e. ( ( A [,] B ) -cn-> CC ) )
47	11	ffvelrnda 5951	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
48	38, 39, 12, 15, 41, 46, 47	ivthicc 21073	. . . . . . . 8 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( ( F ` b ) [,] ( F ` a ) ) C_ ran F )
49	48	adantr 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 )
50	37, 49	eqssd 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 ) )
52	14, 17, 50, 51	syl3anc 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 ) )
53	52	ex 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 ) ) )
54	8, 53	syl5bir 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 ) ) )
55	54	rexlimdvva 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 ) ) )
56	7, 55	mpd 15	1 |- ( ph -> E. x e. RR E. y e. RR ran F = ( x [,] y ) )

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