Theorem iniseg

Description: An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of TakeutiZaring p. 30. (Contributed by NM, 28-Apr-2004)

	Ref	Expression
Assertion	iniseg	⊢ ( 𝐵 ∈ 𝑉 → ( ◡ 𝐴 “ { 𝐵 } ) = { 𝑥 ∣ 𝑥 𝐴 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 ∈ V )
2		vex	⊢ 𝑥 ∈ V
3	2	eliniseg	⊢ ( 𝐵 ∈ V → ( 𝑥 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ 𝑥 𝐴 𝐵 ) )
4	3	abbi2dv	⊢ ( 𝐵 ∈ V → ( ◡ 𝐴 “ { 𝐵 } ) = { 𝑥 ∣ 𝑥 𝐴 𝐵 } )
5	1 4	syl	⊢ ( 𝐵 ∈ 𝑉 → ( ◡ 𝐴 “ { 𝐵 } ) = { 𝑥 ∣ 𝑥 𝐴 𝐵 } )