Theorem uhgrspan1lem1

Description: Lemma 1 for uhgrspan1 . (Contributed by AV, 19-Nov-2020)

	Ref	Expression
Hypotheses	uhgrspan1.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	uhgrspan1.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	uhgrspan1.f	⊢ 𝐹 = { 𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ ( 𝐼 ‘ 𝑖 ) }
Assertion	uhgrspan1lem1	⊢ ( ( 𝑉 ∖ { 𝑁 } ) ∈ V ∧ ( 𝐼 ↾ 𝐹 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		uhgrspan1.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		uhgrspan1.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		uhgrspan1.f	⊢ 𝐹 = { 𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ ( 𝐼 ‘ 𝑖 ) }
4	1	fvexi	⊢ 𝑉 ∈ V
5	4	difexi	⊢ ( 𝑉 ∖ { 𝑁 } ) ∈ V
6	2	fvexi	⊢ 𝐼 ∈ V
7	6	resex	⊢ ( 𝐼 ↾ 𝐹 ) ∈ V
8	5 7	pm3.2i	⊢ ( ( 𝑉 ∖ { 𝑁 } ) ∈ V ∧ ( 𝐼 ↾ 𝐹 ) ∈ V )