mvdco - Metamath Proof Explorer

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Theorem mvdco 17865

Description: Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)

**Assertion**
Ref	Expression
mvdco	$\$ $\$ $\$

**Proof of Theorem mvdco**
Step	Hyp	Ref	Expression
1		inundif 4046	. . . . . . . 8 $\$
2	1	coeq2i 5282	. . . . . . 7 $\$
3		coundi 5636	. . . . . . 7 $\$ $\$
4	2, 3	eqtr3i 2646	. . . . . 6 $\$
5	4	difeq1i 3724	. . . . 5 $\$ $\$ $\$
6		difundir 3880	. . . . 5 $\$ $\$ $\$ $\$ $\$
7	5, 6	eqtri 2644	. . . 4 $\$ $\$ $\$ $\$
8	7	dmeqi 5325	. . 3 $\$ $\$ $\$ $\$
9		dmun 5331	. . 3 $\$ $\$ $\$ $\$ $\$ $\$
10	8, 9	eqtri 2644	. 2 $\$ $\$ $\$ $\$
11		inss2 3834	. . . . . 6
12		coss2 5278	. . . . . 6
13	11, 12	ax-mp 5	. . . . 5
14		cocnvcnv1 5646	. . . . . . 7
15		relcnv 5503	. . . . . . . 8
16		coi1 5651	. . . . . . . 8
17	15, 16	ax-mp 5	. . . . . . 7
18	14, 17	eqtr3i 2646	. . . . . 6
19		cnvcnvss 5589	. . . . . 6
20	18, 19	eqsstri 3635	. . . . 5
21	13, 20	sstri 3612	. . . 4
22		ssdif 3745	. . . 4 $\$ $\$
23		dmss 5323	. . . 4 $\$ $\$ $\$ $\$
24	21, 22, 23	mp2b 10	. . 3 $\$ $\$
25		difss 3737	. . . . 5 $\$ $\$ $\$
26		dmss 5323	. . . . 5 $\$ $\$ $\$ $\$ $\$ $\$
27	25, 26	ax-mp 5	. . . 4 $\$ $\$ $\$
28		dmcoss 5385	. . . 4 $\$ $\$
29	27, 28	sstri 3612	. . 3 $\$ $\$ $\$
30		unss12 3785	. . 3 $\$ $\$ $/\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$ $\$
31	24, 29, 30	mp2an 708	. 2 $\$ $\$ $\$ $\$ $\$
32	10, 31	eqsstri 3635	1 $\$ $\$ $\$

Colors of variables: wff setvar class

Syntax hints:

wceq 1483 $\$ cdif 3571

cun 3572

cin 3573

wss 3574

cid 5023

ccnv 5113

cdm 5114

ccom 5118

wrel 5119

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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126

This theorem is referenced by: f1omvdco2 17868 symgsssg 17887 symgfisg 17888 symggen 17890