Theorem iniseg 4993

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 2746	. 2 |- ( B e. V -> B e. _V )
2		vex 2738	. . . 4 |- x e. _V
3	2	eliniseg 4991	. . 3 |- ( B e. _V -> ( x e. ( `' A " { B } ) <-> x A B ) )
4	3	abbi2dv 2294	. 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 1353   e. wcel 2146   {cab 2161   _Vcvv 2735   {csn 3589   class class class wbr 3998   `'ccnv 4619   "cima 4623

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-cnv 4628  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633

This theorem is referenced by:  dfse2  4994