Theorem eliniseg 3492

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 2462	. . . . 5 |- (x = B -> {x} = {B})
2	1	imaeq2d 3467	. . . 4 |- (x = B -> (`'A"{x}) = (`'A"{B}))
3	2	eleq2d 1578	. . 3 |- (x = B -> (C e. (`'A"{x}) <-> C e. (`'A"{B})))
4		breq2 2673	. . 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 1851	. . . 4 |- x e. V
7		eliniseg.1	. . . 4 |- C e. V
8	6, 7	elimasn 3489	. . 3 |- (C e. (`'A"{x}) <-> <.x, C>. e. `'A)
9		df-br 2670	. . 3 |- (x`'AC <-> <.x, C>. e. `'A)
10	6, 7	brcnv 3363	. . 3 |- (x`'AC <-> CAx)
11	8, 9, 10	3bitr2i 177	. 2 |- (C e. (`'A"{x}) <-> CAx)
12	5, 11	vtoclg 1885	1 |- (B e. D -> (C e. (`'A"{B}) <-> CAB))

Syntax hints:   -> wi 3   <-> wb 144   = wceq 988   e. wcel 990  Vcvv 1849  {csn 2454  <.cop 2456   class class class wbr 2669  `'ccnv 3224  "cima 3228

This theorem is referenced by:  iniseg 3493  isomin 3975  isoini 3976

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-opab 2718  df-xp 3239  df-cnv 3241  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246

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