Definition df-vtx 16009

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 16007	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2813	. . 3 class V
4	2	cv 1397	. . . . 5 class 𝑔
5	3, 3	cxp 4747	. . . . 5 class (V × V)
6	4, 5	wcel 2203	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6332	. . . . 5 class 1st
8	4, 7	cfv 5352	. . . 4 class (1st ‘𝑔)
9		cbs 13212	. . . . 5 class Base
10	4, 9	cfv 5352	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3620	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4171	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1398	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxvalg  16011  1vgrex  16015  wlkreslem  16373  clwwlknonmpo  16423  trlsegvdegfi  16462  eupth2lem3lem1fi  16463  eupth2lem3lem2fi  16464  eupth2lem3lem6fi  16466  eupth2lem3lem4fi  16468