Theorem eliniseg 3421

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 2413	. . . . 5 |- (x = B -> {x} = {B})
2	1	imaeq2d 3396	. . . 4 |- (x = B -> (`'A"{x}) = (`'A"{B}))
3	2	eleq2d 1538	. . 3 |- (x = B -> (C e. (`'A"{x}) <-> C e. (`'A"{B})))
4		breq2 2618	. . 3 |- (x = B -> (CAx <-> CAB))
5	3, 4	bibi12d 628	. 2 |- (x = B -> ((C e. (`'A"{x}) <-> CAx) <-> (C e. (`'A"{B}) <-> CAB)))
6		visset 1809	. . . 4 |- x e. V
7		eliniseg.1	. . . 4 |- C e. V
8	6, 7	elimasn 3418	. . 3 |- (C e. (`'A"{x}) <-> <.x, C>. e. `'A)
9		df-br 2615	. . 3 |- (x`'AC <-> <.x, C>. e. `'A)
10	6, 7	brcnv 3294	. . 3 |- (x`'AC <-> CAx)
11	8, 9, 10	3bitr2 179	. 2 |- (C e. (`'A"{x}) <-> CAx)
12	5, 11	vtoclg 1843	1 |- (B e. D -> (C e. (`'A"{B}) <-> CAB))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956  Vcvv 1807  {csn 2405  <.cop 2407   class class class wbr 2614  `'ccnv 3164  "cima 3168

This theorem is referenced by:  iniseg 3422  isomin 3890  isoini 3891

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186

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