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	$$⊢B\in V\to A^{-1}\left[\left\{B\right\}\right]=\left\{x|xAB\right\}$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢B\in V\to B\in \mathrm{V}$$
2		vex	$$⊢x\in \mathrm{V}$$
3	2	eliniseg	$$⊢B\in \mathrm{V}\to \left(x\in A^{-1}\left[\left\{B\right\}\right]\leftrightarrow xAB\right)$$
4	3	eqabdv	$$⊢B\in \mathrm{V}\to A^{-1}\left[\left\{B\right\}\right]=\left\{x|xAB\right\}$$
5	1 4	syl	$$⊢B\in V\to A^{-1}\left[\left\{B\right\}\right]=\left\{x|xAB\right\}$$