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Metamath Proof Explorer

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 e. V -> ( `' A " { B } ) = { x | x A B } )

Proof

Step Hyp Ref Expression
1 elex 
 |-  ( B e. V -> B e. _V )
2 vex 
 |-  x e. _V
3 2 eliniseg 
 |-  ( B e. _V -> ( x e. ( `' A " { B } ) <-> x A B ) )
4 3 eqabdv 
 |-  ( B e. _V -> ( `' A " { B } ) = { x | x A B } )
5 1 4 syl 
 |-  ( B e. V -> ( `' A " { B } ) = { x | x A B } )