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Theorem iniseg 5238
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 2966 . 2  |-  ( B  e.  V  ->  B  e.  _V )
2 vex 2961 . . . 4  |-  x  e. 
_V
32eliniseg 5236 . . 3  |-  ( B  e.  _V  ->  (
x  e.  ( `' A " { B } )  <->  x A B ) )
43abbi2dv 2553 . 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 1653    e. wcel 1726   {cab 2424   _Vcvv 2958   {csn 3816   class class class wbr 4215   `'ccnv 4880   "cima 4884
This theorem is referenced by:  dffr3  5239  dfse2  5240  dfpred2  25453  inisegn0  27132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894
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