Theorem resieq 2581

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 2066	. . . . . 6 |- (x = C -> (B(I |` A)x <-> B(I |` A)C))
2		cleq2 1110	. . . . . 6 |- (x = C -> (B = x <-> B = C))
3	1, 2	bibi12d 477	. . . . 5 |- (x = C -> ((B(I |` A)x <-> B = x) <-> (B(I |` A)C <-> B = C)))
4	3	imbi2d 464	. . . 4 |- (x = C -> ((B e. A -> (B(I |` A)x <-> B = x)) <-> (B e. A -> (B(I |` A)C <-> B = C))))
5		visset 1350	. . . . . . 7 |- x e. V
6	5	opres 2580	. . . . . 6 |- (B e. A -> (<.B, x>. e. (I |` A) <-> <.B, x>. e. I))
7		ideqg 2126	. . . . . . . 8 |- ((B e. A /\ x e. V) -> (BIx <-> B = x))
8	5, 7	mpan2 519	. . . . . . 7 |- (B e. A -> (BIx <-> B = x))
9		df-br 2063	. . . . . . 7 |- (BIx <-> <.B, x>. e. I)
10	8, 9	syl5bbr 412	. . . . . 6 |- (B e. A -> (<.B, x>. e. I <-> B = x))
11	6, 10	bitrd 406	. . . . 5 |- (B e. A -> (<.B, x>. e. (I |` A) <-> B = x))
12		df-br 2063	. . . . 5 |- (B(I |` A)x <-> <.B, x>. e. (I |` A))
13	11, 12	syl5bb 410	. . . 4 |- (B e. A -> (B(I |` A)x <-> B = x))
14	4, 13	vtoclg 1383	. . 3 |- (C e. A -> (B e. A -> (B(I |` A)C <-> B = C)))
15	14	com12 13	. 2 |- (B e. A -> (C e. A -> (B(I |` A)C <-> B = C)))
16	15	imp 277	1 |- ((B e. A /\ C e. A) -> (B(I |` A)C <-> B = C))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810   class class class wbr 2054  Icid 2057   |` cres 2412

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-res 2430

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