Theorem elinisegOLD 4295

Description: Membership in an initial segment. The idiom (`'A"{B}), meaning {x | xAB}, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30.

Hypothesis

Ref	Expression
eliniseg.1	|- C e. _V

Assertion

Ref	Expression
elinisegOLD	|- (B e. D -> (C e. (`'A"{B}) <-> CAB))

Proof of Theorem elinisegOLD

Step	Hyp	Ref	Expression
1		sneq 3054	. . . . 5 |- (x = B -> {x} = {B})
2	1	imaeq2d 4264	. . . 4 |- (x = B -> (`'A"{x}) = (`'A"{B}))
3	2	eleq2d 1964	. . 3 |- (x = B -> (C e. (`'A"{x}) <-> C e. (`'A"{B})))
4		breq2 3342	. . 3 |- (x = B -> (CAx <-> CAB))
5	3, 4	bibi12d 691	. 2 |- (x = B -> ((C e. (`'A"{x}) <-> CAx) <-> (C e. (`'A"{B}) <-> CAB)))
6		visset 2295	. . . 4 |- x e. _V
7		eliniseg.1	. . . 4 |- C e. _V
8	6, 7	elimasn 4289	. . 3 |- (C e. (`'A"{x}) <-> <.x, C>. e. `'A)
9		df-br 3339	. . 3 |- (x`'AC <-> <.x, C>. e. `'A)
10	6, 7	brcnv 4144	. . 3 |- (x`'AC <-> CAx)
11	8, 9, 10	3bitr2i 196	. 2 |- (C e. (`'A"{x}) <-> CAx)
12	5, 11	vtoclg 2346	1 |- (B e. D -> (C e. (`'A"{B}) <-> CAB))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044  <.cop 3046   class class class wbr 3338  `'ccnv 3985  "cima 3989

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007

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