Definition df-vtx 16255

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 16253	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2821	. . 3 class V
4	2	cv 1401	. . . . 5 class 𝑔
5	3, 3	cxp 4772	. . . . 5 class (V × V)
6	4, 5	wcel 2209	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6372	. . . . 5 class 1st
8	4, 7	cfv 5377	. . . 4 class (1st ‘𝑔)
9		cbs 13352	. . . . 5 class Base
10	4, 9	cfv 5377	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3638	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4192	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1402	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is used by:  vtxvalg  16257  1vgrex  16261  wlkreslem  16619  clwwlknonmpo  16669  trlsegvdegfi  16708  eupth2lem3lem1fi  16709  eupth2lem3lem2fi  16710  eupth2lem3lem6fi  16712  eupth2lem3lem4fi  16714