Theorem evthicc2 20924

Description: Combine ivthicc 20922 with evthicc 20923 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 20923	. . 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 2883	. . 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 2844	. . . 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 20449	. . . . . . . . 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 5839	. . . . . . 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 5839	. . . . . . 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 5554	. . . . . . . . . 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 11352	. . . . . . . . . . . . . 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 5838	. . . . . . . . . . . . . 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 2726	. . . . . . . . . 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 5864	. . . . . . . . 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 5560	. . . . . . . 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 3370	. . . . . . . . . 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 9331	. . . . . . . . . . 11 |- RR C_ CC
43		ssid 3370	. . . . . . . . . . 11 |- CC C_ CC
44		cncfss 20455	. . . . . . . . . . 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 3349	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> F e. ( ( A [,] B ) -cn-> CC ) )
47	11	ffvelrnda 5838	. . . . . . . . 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 20922	. . . . . . . 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 3368	. . . . . 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 6123	. . . . . 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 1218	. . . . 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 2843	. 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 1369   e. wcel 1756   A.wral 2710   E.wrex 2711   C_ wss 3323   class class class wbr 4287   ran crn 4836   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   <_ cle 9411   [,]cicc 11295   -cn->ccncf 20432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cn 18811  df-cnp 18812  df-cmp 18970  df-tx 19115  df-hmeo 19308  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434

This theorem is referenced by:  dvcnvrelem1  21469