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Theorem iniseg 4747
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 2619 . 2  |-  ( B  e.  V  ->  B  e.  _V )
2 vex 2613 . . . 4  |-  x  e. 
_V
32eliniseg 4745 . . 3  |-  ( B  e.  _V  ->  (
x  e.  ( `' A " { B } )  <->  x A B ) )
43abbi2dv 2201 . 2  |-  ( B  e.  _V  ->  ( `' A " { B } )  =  {
x  |  x A B } )
51, 4syl 14 1  |-  ( B  e.  V  ->  ( `' A " { B } )  =  {
x  |  x A B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   {cab 2069   _Vcvv 2610   {csn 3416   class class class wbr 3805   `'ccnv 4390   "cima 4394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404
This theorem is referenced by:  dfse2  4748
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