Theorem evthicc2 23229

Description: Combine ivthicc 23227 with evthicc 23228 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 23228	. . 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 3107	. . 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 224	. 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 3064	. . . 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 481	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> F e. ( ( A [,] B ) -cn-> RR ) )
10		cncff 22696	. . . . . . . . 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 796	. . . . . . . 8 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> b e. ( A [,] B ) )
13	11, 12	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( F ` b ) e. RR )
14	13	adantr 481	. . . . . 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 794	. . . . . . . 8 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> a e. ( A [,] B ) )
16	11, 15	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( F ` a ) e. RR )
17	16	adantr 481	. . . . . 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 481	. . . . . . . . . 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 6045	. . . . . . . . . 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 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` b ) e. RR )
22	16	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` a ) e. RR )
23		elicc2 12238	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 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 1042	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 466	. . . . . . . . . . . . 13 |- ( ( ( F ` z ) <_ ( F ` a ) /\ ( F ` b ) <_ ( F ` z ) ) <-> ( ( F ` b ) <_ ( F ` z ) /\ ( F ` z ) <_ ( F ` a ) ) )
28	11	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` z ) e. RR )
29	28	biantrurd 529	. . . . . . . . . . . . 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 272	. . . . . . . . . . . 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 271	. . . . . . . . . . 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 2985	. . . . . . . . . 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 502	. . . . . . . . 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 6388	. . . . . . . . 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 698	. . . . . . . 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 6053	. . . . . . . 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 481	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> A e. RR )
39	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> B e. RR )
40		ssid 3624	. . . . . . . . . 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 9993	. . . . . . . . . . 11 |- RR C_ CC
43		ssid 3624	. . . . . . . . . . 11 |- CC C_ CC
44		cncfss 22702	. . . . . . . . . . 11 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC ) )
45	42, 43, 44	mp2an 708	. . . . . . . . . 10 |- ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC )
46	45, 9	sseldi 3601	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> F e. ( ( A [,] B ) -cn-> CC ) )
47	11	ffvelrnda 6359	. . . . . . . . 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 23227	. . . . . . . 8 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( ( F ` b ) [,] ( F ` a ) ) C_ ran F )
49	48	adantr 481	. . . . . . 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 3620	. . . . . 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 6692	. . . . . 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 1326	. . . . 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 450	. . . 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 233	. . 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 3038	. 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   <_ cle 10075   [,]cicc 12178   -cn->ccncf 22679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681

This theorem is referenced by:  dvcnvrelem1  23780