Theorem isuvtx

Description: The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 30-Oct-2020) (Revised by AV, 14-Feb-2022)

	Ref	Expression
Hypotheses	uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isuvtx.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	isuvtx	⊢ ( UnivVtx ‘ 𝐺 ) = { 𝑣 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 }

Proof

Step	Hyp	Ref	Expression
1		uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isuvtx.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3	1	uvtxval	⊢ ( UnivVtx ‘ 𝐺 ) = { 𝑣 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) }
4	1 2	nbgrel	⊢ ( 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) )
5		df-3an	⊢ ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) ↔ ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) )
6	4 5	bitri	⊢ ( 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) )
7		prcom	⊢ { 𝑘 , 𝑣 } = { 𝑣 , 𝑘 }
8	7	sseq1i	⊢ ( { 𝑘 , 𝑣 } ⊆ 𝑒 ↔ { 𝑣 , 𝑘 } ⊆ 𝑒 )
9	8	rexbii	⊢ ( ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 )
10		id	⊢ ( 𝑣 ∈ 𝑉 → 𝑣 ∈ 𝑉 )
11		eldifi	⊢ ( 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) → 𝑘 ∈ 𝑉 )
12	10 11	anim12ci	⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) )
13		eldifsni	⊢ ( 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) → 𝑘 ≠ 𝑣 )
14	13	adantl	⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → 𝑘 ≠ 𝑣 )
15	12 14	jca	⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) )
16	15	biantrurd	⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ↔ ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) ) )
17	9 16	syl5rbb	⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) ↔ ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 ) )
18	6 17	syl5bb	⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 ) )
19	18	ralbidva	⊢ ( 𝑣 ∈ 𝑉 → ( ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 ) )
20	19	rabbiia	⊢ { 𝑣 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) } = { 𝑣 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 }
21	3 20	eqtri	⊢ ( UnivVtx ‘ 𝐺 ) = { 𝑣 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 }