Theorem edgssv2

Description: An edge of a simple graph is an unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020) (Revised by AV, 23-Oct-2020)

	Ref	Expression
Hypotheses	edgssv2.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	edgssv2.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	edgssv2	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ⊆ 𝑉 ∧ ( ♯ ‘ 𝐶 ) = 2 ) )

Proof

Step	Hyp	Ref	Expression
1		edgssv2.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		edgssv2.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3	2	eleq2i	⊢ ( 𝐶 ∈ 𝐸 ↔ 𝐶 ∈ ( Edg ‘ 𝐺 ) )
4		edgusgr	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ ( Edg ‘ 𝐺 ) ) → ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
5	3 4	sylan2b	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
6		elpwi	⊢ ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) → 𝐶 ⊆ ( Vtx ‘ 𝐺 ) )
7	6	anim1i	⊢ ( ( 𝐶 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) → ( 𝐶 ⊆ ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
8	5 7	syl	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ⊆ ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) )
9	1	a1i	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → 𝑉 = ( Vtx ‘ 𝐺 ) )
10	9	sseq2d	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ⊆ 𝑉 ↔ 𝐶 ⊆ ( Vtx ‘ 𝐺 ) ) )
11	10	anbi1d	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( ( 𝐶 ⊆ 𝑉 ∧ ( ♯ ‘ 𝐶 ) = 2 ) ↔ ( 𝐶 ⊆ ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐶 ) = 2 ) ) )
12	8 11	mpbird	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ 𝐸 ) → ( 𝐶 ⊆ 𝑉 ∧ ( ♯ ‘ 𝐶 ) = 2 ) )