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Theorem iniseg 5226
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
StepHypRef Expression
1 elex 2956 . 2  |-  ( B  e.  V  ->  B  e.  _V )
2 vex 2951 . . . 4  |-  x  e. 
_V
32eliniseg 5224 . . 3  |-  ( B  e.  _V  ->  (
x  e.  ( `' A " { B } )  <->  x A B ) )
43abbi2dv 2550 . 2  |-  ( B  e.  _V  ->  ( `' A " { B } )  =  {
x  |  x A B } )
51, 4syl 16 1  |-  ( B  e.  V  ->  ( `' A " { B } )  =  {
x  |  x A B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948   {csn 3806   class class class wbr 4204   `'ccnv 4868   "cima 4872
This theorem is referenced by:  dffr3  5227  dfse2  5228  dfpred2  25430  inisegn0  27055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882
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