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Theorem iniseg 5198
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 2928 . 2  |-  ( B  e.  V  ->  B  e.  _V )
2 vex 2923 . . . 4  |-  x  e. 
_V
32eliniseg 5196 . . 3  |-  ( B  e.  _V  ->  (
x  e.  ( `' A " { B } )  <->  x A B ) )
43abbi2dv 2523 . 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 1649    e. wcel 1721   {cab 2394   _Vcvv 2920   {csn 3778   class class class wbr 4176   `'ccnv 4840   "cima 4844
This theorem is referenced by:  dffr3  5199  dfse2  5200  dfpred2  25393  inisegn0  27012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-xp 4847  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854
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