Definition df-eupth 16367

Description: Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

Assertion

Ref	Expression
df-eupth	|- EulerPaths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ f : ( 0..^ (♯ ` f ) ) -onto-> dom (iEdg ` g ) ) } )

Distinct variable group:   f, g, p

Detailed syntax breakdown of Definition df-eupth

Step	Hyp	Ref	Expression
1		ceupth 16366	. 2 class EulerPaths
2		vg	. . 3 setvar g
3		cvv 2803	. . 3 class _V
4		vf	. . . . . . 7 setvar f
5	4	cv 1397	. . . . . 6 class f
6		vp	. . . . . . 7 setvar p
7	6	cv 1397	. . . . . 6 class p
8	2	cv 1397	. . . . . . 7 class g
9		ctrls 16304	. . . . . . 7 class Trails
10	8, 9	cfv 5333	. . . . . 6 class (Trails ` g )
11	5, 7, 10	wbr 4093	. . . . 5 wff f (Trails ` g ) p
12		cc0 8075	. . . . . . 7 class 0
13		chash 11083	. . . . . . . 8 class ♯
14	5, 13	cfv 5333	. . . . . . 7 class (♯ ` f )
15		cfzo 10422	. . . . . . 7 class ..^
16	12, 14, 15	co 6028	. . . . . 6 class ( 0..^ (♯ ` f ) )
17		ciedg 15937	. . . . . . . 8 class iEdg
18	8, 17	cfv 5333	. . . . . . 7 class (iEdg ` g )
19	18	cdm 4731	. . . . . 6 class dom (iEdg ` g )
20	16, 19, 5	wfo 5331	. . . . 5 wff f : ( 0..^ (♯ ` f ) ) -onto-> dom (iEdg ` g )
21	11, 20	wa 104	. . . 4 wff ( f (Trails ` g ) p /\ f : ( 0..^ (♯ ` f ) ) -onto-> dom (iEdg ` g ) )
22	21, 4, 6	copab 4154	. . 3 class { <. f , p >. | ( f (Trails ` g ) p /\ f : ( 0..^ (♯ ` f ) ) -onto-> dom (iEdg ` g ) ) }
23	2, 3, 22	cmpt 4155	. 2 class ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ f : ( 0..^ (♯ ` f ) ) -onto-> dom (iEdg ` g ) ) } )
24	1, 23	wceq 1398	1 wff EulerPaths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ f : ( 0..^ (♯ ` f ) ) -onto-> dom (iEdg ` g ) ) } )

This definition is referenced by:  releupth  16368  eupthsg  16369  eupthv  16370