Theorem upgrres1lem1

Description: Lemma 1 for upgrres1 . (Contributed by AV, 7-Nov-2020)

	Ref	Expression
Hypotheses	upgrres1.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	upgrres1.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
	upgrres1.f	⊢ 𝐹 = { 𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒 }
Assertion	upgrres1lem1	⊢ ( ( 𝑉 ∖ { 𝑁 } ) ∈ V ∧ ( I ↾ 𝐹 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		upgrres1.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		upgrres1.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		upgrres1.f	⊢ 𝐹 = { 𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒 }
4	1	fvexi	⊢ 𝑉 ∈ V
5	4	difexi	⊢ ( 𝑉 ∖ { 𝑁 } ) ∈ V
6	2	fvexi	⊢ 𝐸 ∈ V
7	3 6	rabex2	⊢ 𝐹 ∈ V
8		resiexg	⊢ ( 𝐹 ∈ V → ( I ↾ 𝐹 ) ∈ V )
9	7 8	ax-mp	⊢ ( I ↾ 𝐹 ) ∈ V
10	5 9	pm3.2i	⊢ ( ( 𝑉 ∖ { 𝑁 } ) ∈ V ∧ ( I ↾ 𝐹 ) ∈ V )