Theorem evthicc2 19310

Description: Combine ivthicc 19308 with evthicc 19309 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 19309	. . 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 2835	. . 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 204	. 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 2798	. . . 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 452	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> F e. ( ( A [,] B ) -cn-> RR ) )
10		cncff 18876	. . . . . . . . 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 734	. . . . . . . 8 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> b e. ( A [,] B ) )
13	11, 12	ffvelrnd 5830	. . . . . . 7 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( F ` b ) e. RR )
14	13	adantr 452	. . . . . 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 733	. . . . . . . 8 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> a e. ( A [,] B ) )
16	11, 15	ffvelrnd 5830	. . . . . . 7 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( F ` a ) e. RR )
17	16	adantr 452	. . . . . 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 452	. . . . . . . . . 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 5550	. . . . . . . . . 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 452	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` b ) e. RR )
22	16	adantr 452	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` a ) e. RR )
23		elicc2 10931	. . . . . . . . . . . . . 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 643	. . . . . . . . . . . . 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 940	. . . . . . . . . . . . 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 253	. . . . . . . . . . . 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 438	. . . . . . . . . . . . 13 |- ( ( ( F ` z ) <_ ( F ` a ) /\ ( F ` b ) <_ ( F ` z ) ) <-> ( ( F ` b ) <_ ( F ` z ) /\ ( F ` z ) <_ ( F ` a ) ) )
28	11	ffvelrnda 5829	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) /\ z e. ( A [,] B ) ) -> ( F ` z ) e. RR )
29	28	biantrurd 495	. . . . . . . . . . . . 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 249	. . . . . . . . . . . 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 248	. . . . . . . . . . 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 2682	. . . . . . . . . 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 472	. . . . . . . . 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 5853	. . . . . . . . 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 646	. . . . . . . 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 5556	. . . . . . . 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 452	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> A e. RR )
39	2	adantr 452	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> B e. RR )
40		ssid 3327	. . . . . . . . . 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 9003	. . . . . . . . . . 11 |- RR C_ CC
43		ssid 3327	. . . . . . . . . . 11 |- CC C_ CC
44		cncfss 18882	. . . . . . . . . . 11 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC ) )
45	42, 43, 44	mp2an 654	. . . . . . . . . 10 |- ( ( A [,] B ) -cn-> RR ) C_ ( ( A [,] B ) -cn-> CC )
46	45, 9	sseldi 3306	. . . . . . . . 9 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> F e. ( ( A [,] B ) -cn-> CC ) )
47	11	ffvelrnda 5829	. . . . . . . . 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 19308	. . . . . . . 8 |- ( ( ph /\ ( a e. ( A [,] B ) /\ b e. ( A [,] B ) ) ) -> ( ( F ` b ) [,] ( F ` a ) ) C_ ran F )
49	48	adantr 452	. . . . . . 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 3325	. . . . . 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 6075	. . . . . 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 1184	. . . . 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 424	. . . 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 210	. . 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 2797	. 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 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   C_ wss 3280   class class class wbr 4172   ran crn 4838   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   <_ cle 9077   [,]cicc 10875   -cn->ccncf 18859

This theorem is referenced by:  dvcnvrelem1  19854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861