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.

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 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 ) ) )
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 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 ) ) )
9	4	adantr 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 )
11	9, 10	syl 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 ) )
13	11, 12	ffvelrnd 6009	. . . . . . 7 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( F ` b ) e. RR )
14	13	adantr 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 ) )
16	11, 15	ffvelrnd 6009	. . . . . . 7 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( F ` a ) e. RR )
17	16	adantr 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 )
18	11	adantr 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 ) )
20	18, 19	syl 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 ) )
21	13	adantr 463	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` b ) e. RR )
22	16	adantr 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 ) ) ) )
24	21, 22, 23	syl2anc 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 ) ) ) )
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 448	. . . . . . . . . . . . 13 |- ( ( ( F ` z ) <_ ( F ` a ) /\ ( F ` b ) <_ ( F ` z ) ) <-> ( ( F ` b ) <_ ( F ` z ) /\ ( F ` z ) <_ ( F ` a ) ) )
28	11	ffvelrnda 6008	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` z ) e. RR )
29	28	biantrurd 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 ) ) ) ) )
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 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 ) ) ) )
33	32	biimpar 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 ) ) ) )
35	20, 33, 34	sylanbrc 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 ) ) )
37	35, 36	syl 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 ) ) )
38	1	adantr 463	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> A e. RR )
39	2	adantr 463	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> B e. RR )
40		ssid 3460	. . . . . . . . . 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 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 ) )
45	42, 43, 44	mp2an 670	. . . . . . . . . 10 |- ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC )
46	45, 9	sseldi 3439	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> F e. ( ( A [,] B ) -cn-> CC ) )
47	11	ffvelrnda 6008	. . . . . . . . 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 22160	. . . . . . . 8 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( ( F ` b ) [,] ( F ` a ) ) C_ ran F )
49	48	adantr 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 )
50	37, 49	eqssd 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 ) )
52	14, 17, 50, 51	syl3anc 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 ) )
53	52	ex 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 ) ) )
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 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 ) ) )
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 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