Theorem iniseg 5073

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 2788	. 2 |- ( B e. V -> B e. _V )
2		vex 2779	. . . 4 |- x e. _V
3	2	eliniseg 5071	. . 3 |- ( B e. _V -> ( x e. ( `' A " { B } ) <-> x A B ) )
4	3	abbi2dv 2326	. 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 1373   e. wcel 2178   {cab 2193   _Vcvv 2776   {csn 3643   class class class wbr 4059   `'ccnv 4692   "cima 4696

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  dfse2  5074