Theorem usgrss

Description: An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017) (Revised by AV, 15-Oct-2020)

	Ref	Expression
Hypotheses	usgrf1o.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
	usgrss.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	usgrss	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( 𝐸 ‘ 𝑋 ) ⊆ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		usgrf1o.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2		usgrss.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
3		ssrab2	⊢ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ⊆ 𝒫 𝑉
4	2 1	usgrfs	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } )
5		f1f	⊢ ( 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } → 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } )
6	4 5	syl	⊢ ( 𝐺 ∈ USGraph → 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } )
7	6	ffvelrnda	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( 𝐸 ‘ 𝑋 ) ∈ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } )
8	3 7	sseldi	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( 𝐸 ‘ 𝑋 ) ∈ 𝒫 𝑉 )
9	8	elpwid	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸 ) → ( 𝐸 ‘ 𝑋 ) ⊆ 𝑉 )