Theorem eliniseg 3435

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

Proof of Theorem eliniseg

Step	Hyp	Ref	Expression
1		sneq 2421	. . . . 5 |- (x = B -> {x} = {B})
2	1	imaeq2d 3410	. . . 4 |- (x = B -> (`'A"{x}) = (`'A"{B}))
3	2	eleq2d 1544	. . 3 |- (x = B -> (C e. (`'A"{x}) <-> C e. (`'A"{B})))
4		breq2 2628	. . 3 |- (x = B -> (CAx <-> CAB))
5	3, 4	bibi12d 631	. 2 |- (x = B -> ((C e. (`'A"{x}) <-> CAx) <-> (C e. (`'A"{B}) <-> CAB)))
6		visset 1816	. . . 4 |- x e. V
7		eliniseg.1	. . . 4 |- C e. V
8	6, 7	elimasn 3432	. . 3 |- (C e. (`'A"{x}) <-> <.x, C>. e. `'A)
9		df-br 2625	. . 3 |- (x`'AC <-> <.x, C>. e. `'A)
10	6, 7	brcnv 3305	. . 3 |- (x`'AC <-> CAx)
11	8, 9, 10	3bitr2 179	. 2 |- (C e. (`'A"{x}) <-> CAx)
12	5, 11	vtoclg 1850	1 |- (B e. D -> (C e. (`'A"{B}) <-> CAB))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  Vcvv 1814  {csn 2413  <.cop 2415   class class class wbr 2624  `'ccnv 3175  "cima 3179

This theorem is referenced by:  iniseg 3436  isomin 3905  isoini 3906

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197

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