Theorem upgrres1lem2

Description: Lemma 2 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 }
	upgrres1.s	⊢ 𝑆 = ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( I ↾ 𝐹 ) ⟩
Assertion	upgrres1lem2	⊢ ( Vtx ‘ 𝑆 ) = ( 𝑉 ∖ { 𝑁 } )

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		upgrres1.s	⊢ 𝑆 = ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( I ↾ 𝐹 ) ⟩
5	4	fveq2i	⊢ ( Vtx ‘ 𝑆 ) = ( Vtx ‘ ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( I ↾ 𝐹 ) ⟩ )
6	1 2 3	upgrres1lem1	⊢ ( ( 𝑉 ∖ { 𝑁 } ) ∈ V ∧ ( I ↾ 𝐹 ) ∈ V )
7		opvtxfv	⊢ ( ( ( 𝑉 ∖ { 𝑁 } ) ∈ V ∧ ( I ↾ 𝐹 ) ∈ V ) → ( Vtx ‘ ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( I ↾ 𝐹 ) ⟩ ) = ( 𝑉 ∖ { 𝑁 } ) )
8	6 7	ax-mp	⊢ ( Vtx ‘ ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( I ↾ 𝐹 ) ⟩ ) = ( 𝑉 ∖ { 𝑁 } )
9	5 8	eqtri	⊢ ( Vtx ‘ 𝑆 ) = ( 𝑉 ∖ { 𝑁 } )