Theorem uvtxel1

Description: Characterization of a universal vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 14-Feb-2022)

	Ref	Expression
Hypotheses	uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isuvtx.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	uvtxel1	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝑁 ∈ 𝑉 ∧ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑁 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑁 } ⊆ 𝑒 ) )

Proof

Step	Hyp	Ref	Expression
1		uvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isuvtx.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		sneq	⊢ ( 𝑛 = 𝑁 → { 𝑛 } = { 𝑁 } )
4	3	difeq2d	⊢ ( 𝑛 = 𝑁 → ( 𝑉 ∖ { 𝑛 } ) = ( 𝑉 ∖ { 𝑁 } ) )
5		preq2	⊢ ( 𝑛 = 𝑁 → { 𝑘 , 𝑛 } = { 𝑘 , 𝑁 } )
6	5	sseq1d	⊢ ( 𝑛 = 𝑁 → ( { 𝑘 , 𝑛 } ⊆ 𝑒 ↔ { 𝑘 , 𝑁 } ⊆ 𝑒 ) )
7	6	rexbidv	⊢ ( 𝑛 = 𝑁 → ( ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑛 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑁 } ⊆ 𝑒 ) )
8	4 7	raleqbidv	⊢ ( 𝑛 = 𝑁 → ( ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑛 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑛 } ⊆ 𝑒 ↔ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑁 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑁 } ⊆ 𝑒 ) )
9	1 2	isuvtx	⊢ ( UnivVtx ‘ 𝐺 ) = { 𝑛 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑛 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑛 } ⊆ 𝑒 }
10	8 9	elrab2	⊢ ( 𝑁 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( 𝑁 ∈ 𝑉 ∧ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑁 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑁 } ⊆ 𝑒 ) )