Theorem eliniseg 3429

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 2417	. . . . 5 |- (x = B -> {x} = {B})
2	1	imaeq2d 3404	. . . 4 |- (x = B -> (`'A"{x}) = (`'A"{B}))
3	2	eleq2d 1541	. . 3 |- (x = B -> (C e. (`'A"{x}) <-> C e. (`'A"{B})))
4		breq2 2623	. . 3 |- (x = B -> (CAx <-> CAB))
5	3, 4	bibi12d 629	. 2 |- (x = B -> ((C e. (`'A"{x}) <-> CAx) <-> (C e. (`'A"{B}) <-> CAB)))
6		visset 1813	. . . 4 |- x e. V
7		eliniseg.1	. . . 4 |- C e. V
8	6, 7	elimasn 3426	. . 3 |- (C e. (`'A"{x}) <-> <.x, C>. e. `'A)
9		df-br 2620	. . 3 |- (x`'AC <-> <.x, C>. e. `'A)
10	6, 7	brcnv 3299	. . 3 |- (x`'AC <-> CAx)
11	8, 9, 10	3bitr2 179	. 2 |- (C e. (`'A"{x}) <-> CAx)
12	5, 11	vtoclg 1847	1 |- (B e. D -> (C e. (`'A"{B}) <-> CAB))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409  <.cop 2411   class class class wbr 2619  `'ccnv 3169  "cima 3173

This theorem is referenced by:  iniseg 3430  isomin 3899  isoini 3900

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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