Theorem resieq 3360

Description: A restricted identity relation is equivalent to equality in its domain.

Assertion

Ref	Expression
resieq	|- ((B e. A /\ C e. A) -> (B(I |` A)C <-> B = C))

Proof of Theorem resieq

Step	Hyp	Ref	Expression
1		breq2 2613	. . . . 5 |- (x = C -> (B(I |` A)x <-> B(I |` A)C))
2		eqeq2 1476	. . . . 5 |- (x = C -> (B = x <-> B = C))
3	1, 2	bibi12d 627	. . . 4 |- (x = C -> ((B(I |` A)x <-> B = x) <-> (B(I |` A)C <-> B = C)))
4	3	imbi2d 610	. . 3 |- (x = C -> ((B e. A -> (B(I |` A)x <-> B = x)) <-> (B e. A -> (B(I |` A)C <-> B = C))))
5		visset 1804	. . . . 5 |- x e. V
6	5	opres 3359	. . . 4 |- (B e. A -> (<.B, x>. e. (I |` A) <-> <.B, x>. e. I))
7		df-br 2610	. . . 4 |- (B(I |` A)x <-> <.B, x>. e. (I |` A))
8	5	ideq 3267	. . . . 5 |- (BIx <-> B = x)
9		df-br 2610	. . . . 5 |- (BIx <-> <.B, x>. e. I)
10	8, 9	bitr3 175	. . . 4 |- (B = x <-> <.B, x>. e. I)
11	6, 7, 10	3bitr4g 553	. . 3 |- (B e. A -> (B(I |` A)x <-> B = x))
12	4, 11	vtoclg 1838	. 2 |- (C e. A -> (B e. A -> (B(I |` A)C <-> B = C)))
13	12	impcom 351	1 |- ((B e. A /\ C e. A) -> (B(I |` A)C <-> B = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  <.cop 2401   class class class wbr 2609  Icid 2820   |` cres 3162

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-res 3180

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