Theorem iniseg 3430

Description: An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30.

Assertion

Ref	Expression
iniseg	|- (B e. C -> (`'A"{B}) = {x | xAB})

Distinct variable groups:   x,A   x,B

Proof of Theorem iniseg

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (B e. C -> B e. V)
2		visset 1813	. . . 4 |- x e. V
3	2	eliniseg 3429	. . 3 |- (B e. V -> (x e. (`'A"{B}) <-> xAB))
4	3	abbi2dv 1578	. 2 |- (B e. V -> (`'A"{B}) = {x | xAB})
5	1, 4	syl 10	1 |- (B e. C -> (`'A"{B}) = {x | xAB})

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811  {csn 2409   class class class wbr 2619  `'ccnv 3169  "cima 3173

This theorem is referenced by:  dffr3 3431

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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