Theorem iniseg 5042

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 2774	. 2 |- ( B e. V -> B e. _V )
2		vex 2766	. . . 4 |- x e. _V
3	2	eliniseg 5040	. . 3 |- ( B e. _V -> ( x e. ( `' A " { B } ) <-> x A B ) )
4	3	abbi2dv 2315	. 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 1364   e. wcel 2167   {cab 2182   _Vcvv 2763   {csn 3623   class class class wbr 4034   `'ccnv 4663   "cima 4667

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677

This theorem is referenced by:  dfse2  5043