Theorem iniseg 5154

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.)

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem iniseg

Step	Hyp	Ref	Expression
1		elex 2833	. 2 |- ( B e. V -> B e. _V )
2		vex 2824	. . . 4 |- x e. _V
3	2	eliniseg 5152	. . 3 |- ( B e. _V -> ( x e. ( `' A " { B } ) <-> x A B ) )
4	3	abbi2dv 2359	. 2 |- ( B e. _V -> ( `' A " { B } ) = { x | x A B } )
5	1, 4	syl 14	1 |- ( B e. V -> ( `' A " { B } ) = { x | x A B } )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   {csn 3705   class class class wbr 4125   `'ccnv 4768   "cima 4772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by:  dfse2  5155