Theorem uhgrspan1lem3

Description: Lemma 3 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 ) }
	uhgrspan1.s	⊢ 𝑆 = ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( 𝐼 ↾ 𝐹 ) ⟩
Assertion	uhgrspan1lem3	⊢ ( iEdg ‘ 𝑆 ) = ( 𝐼 ↾ 𝐹 )

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		uhgrspan1.s	⊢ 𝑆 = ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( 𝐼 ↾ 𝐹 ) ⟩
5	4	fveq2i	⊢ ( iEdg ‘ 𝑆 ) = ( iEdg ‘ ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( 𝐼 ↾ 𝐹 ) ⟩ )
6	1 2 3	uhgrspan1lem1	⊢ ( ( 𝑉 ∖ { 𝑁 } ) ∈ V ∧ ( 𝐼 ↾ 𝐹 ) ∈ V )
7		opiedgfv	⊢ ( ( ( 𝑉 ∖ { 𝑁 } ) ∈ V ∧ ( 𝐼 ↾ 𝐹 ) ∈ V ) → ( iEdg ‘ ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( 𝐼 ↾ 𝐹 ) ⟩ ) = ( 𝐼 ↾ 𝐹 ) )
8	6 7	ax-mp	⊢ ( iEdg ‘ ⟨ ( 𝑉 ∖ { 𝑁 } ) , ( 𝐼 ↾ 𝐹 ) ⟩ ) = ( 𝐼 ↾ 𝐹 )
9	5 8	eqtri	⊢ ( iEdg ‘ 𝑆 ) = ( 𝐼 ↾ 𝐹 )