ovolficc - Metamath Proof Explorer

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Theorem ovolficc 23237

Description: Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)

**Assertion**
Ref	Expression
ovolficc	$/\$ $/\$

Distinct variable groups:

**Proof of Theorem ovolficc**
Step	Hyp	Ref	Expression
1		iccf 12272	. . . . . 6
2		inss2 3834	. . . . . . . 8
3		rexpssxrxp 10084	. . . . . . . 8
4	2, 3	sstri 3612	. . . . . . 7
5		fss 6056	. . . . . . 7 $/\$
6	4, 5	mpan2 707	. . . . . 6
7		fco 6058	. . . . . 6 $/\$
8	1, 6, 7	sylancr 695	. . . . 5
9		ffn 6045	. . . . 5
10		fniunfv 6505	. . . . 5
11	8, 9, 10	3syl 18	. . . 4
12	11	sseq2d 3633	. . 3
13	12	adantl 482	. 2 $/\$
14		dfss3 3592	. . 3
15		ssel2 3598	. . . . . 6 $/\$
16		eliun 4524	. . . . . . 7
17		fvco3 6275	. . . . . . . . . . . . 13 $/\$
18		ffvelrn 6357	. . . . . . . . . . . . . . . . 17 $/\$
19	2, 18	sseldi 3601	. . . . . . . . . . . . . . . 16 $/\$
20		1st2nd2 7205	. . . . . . . . . . . . . . . 16
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 $/\$
22	21	fveq2d 6195	. . . . . . . . . . . . . 14 $/\$
23		df-ov 6653	. . . . . . . . . . . . . 14
24	22, 23	syl6eqr 2674	. . . . . . . . . . . . 13 $/\$
25	17, 24	eqtrd 2656	. . . . . . . . . . . 12 $/\$
26	25	eleq2d 2687	. . . . . . . . . . 11 $/\$
27		ovolfcl 23235	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
28		elicc2 12238	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
29		3anass 1042	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
30	28, 29	syl6bb 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
31	30	3adant3 1081	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
32	27, 31	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$
33	26, 32	bitrd 268	. . . . . . . . . 10 $/\$ $/\$ $/\$
34	33	adantll 750	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35		simpll 790	. . . . . . . . . 10 $/\$ $/\$
36	35	biantrurd 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
37	34, 36	bitr4d 271	. . . . . . . 8 $/\$ $/\$ $/\$
38	37	rexbidva 3049	. . . . . . 7 $/\$ $/\$
39	16, 38	syl5bb 272	. . . . . 6 $/\$ $/\$
40	15, 39	sylan 488	. . . . 5 $/\$ $/\$ $/\$
41	40	an32s 846	. . . 4 $/\$ $/\$ $/\$
42	41	ralbidva 2985	. . 3 $/\$ $/\$
43	14, 42	syl5bb 272	. 2 $/\$ $/\$
44	13, 43	bitr3d 270	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

wrex 2913

cin 3573

wss 3574

cpw 4158

cop 4183

cuni 4436

ciun 4520 class class class wbr 4653

cxp 5112

crn 5115

ccom 5118

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

cr 9935

cxr 10073

cle 10075

cn 11020

cicc 12178

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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011

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-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-icc 12182

This theorem is referenced by: ovollb2lem 23256 ovolctb 23258 ovolicc1 23284 ioombl1lem4 23329

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