Definition df-vtx 15884

Description: Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)

Assertion

Ref	Expression
df-vtx	⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

Detailed syntax breakdown of Definition df-vtx

Step	Hyp	Ref	Expression
1		cvtx 15882	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2802	. . 3 class V
4	2	cv 1396	. . . . 5 class 𝑔
5	3, 3	cxp 4723	. . . . 5 class (V × V)
6	4, 5	wcel 2202	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6301	. . . . 5 class 1st
8	4, 7	cfv 5326	. . . 4 class (1st ‘𝑔)
9		cbs 13100	. . . . 5 class Base
10	4, 9	cfv 5326	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3605	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4150	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1397	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxvalg  15886  1vgrex  15890  wlkreslem  16248  clwwlknonmpo  16298  trlsegvdegfi  16337  eupth2lem3lem1fi  16338  eupth2lem3lem2fi  16339  eupth2lem3lem6fi  16341  eupth2lem3lem4fi  16343