Definition df-vtx

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)

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

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cvtx	⊢ Vtx
1		vg	⊢ 𝑔
2		cvv	⊢ V
3	1	cv	⊢ 𝑔
4	2 2	cxp	⊢ ( V × V )
5	3 4	wcel	⊢ 𝑔 ∈ ( V × V )
6		c1st	⊢ 1st
7	3 6	cfv	⊢ ( 1st ‘ 𝑔 )
8		cbs	⊢ Base
9	3 8	cfv	⊢ ( Base ‘ 𝑔 )
10	5 7 9	cif	⊢ if ( 𝑔 ∈ ( V × V ) , ( 1st ‘ 𝑔 ) , ( Base ‘ 𝑔 ) )
11	1 2 10	cmpt	⊢ ( 𝑔 ∈ V ↦ if ( 𝑔 ∈ ( V × V ) , ( 1st ‘ 𝑔 ) , ( Base ‘ 𝑔 ) ) )
12	0 11	wceq	⊢ Vtx = ( 𝑔 ∈ V ↦ if ( 𝑔 ∈ ( V × V ) , ( 1st ‘ 𝑔 ) , ( Base ‘ 𝑔 ) ) )