Theorem resieq 2581

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

Assertion

Ref	Expression
resieq	⊢ ((B ∈ A ∧ C ∈ 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 ∈ A → (B(I ↾ A)x ↔ B = x)) ↔ (B ∈ A → (B(I ↾ A)C ↔ B = C))))
5		visset 1350	. . . . . . 7 ⊢ x ∈ V
6	5	opres 2580	. . . . . 6 ⊢ (B ∈ A → (⟨B, x⟩ ∈ (I ↾ A) ↔ ⟨B, x⟩ ∈ I))
7		ideqg 2126	. . . . . . . 8 ⊢ ((B ∈ A ∧ x ∈ V) → (BIx ↔ B = x))
8	5, 7	mpan2 519	. . . . . . 7 ⊢ (B ∈ A → (BIx ↔ B = x))
9		df-br 2063	. . . . . . 7 ⊢ (BIx ↔ ⟨B, x⟩ ∈ I)
10	8, 9	syl5bbr 412	. . . . . 6 ⊢ (B ∈ A → (⟨B, x⟩ ∈ I ↔ B = x))
11	6, 10	bitrd 406	. . . . 5 ⊢ (B ∈ A → (⟨B, x⟩ ∈ (I ↾ A) ↔ B = x))
12		df-br 2063	. . . . 5 ⊢ (B(I ↾ A)x ↔ ⟨B, x⟩ ∈ (I ↾ A))
13	11, 12	syl5bb 410	. . . 4 ⊢ (B ∈ A → (B(I ↾ A)x ↔ B = x))
14	4, 13	vtoclg 1383	. . 3 ⊢ (C ∈ A → (B ∈ A → (B(I ↾ A)C ↔ B = C)))
15	14	com12 13	. 2 ⊢ (B ∈ A → (C ∈ A → (B(I ↾ A)C ↔ B = C)))
16	15	imp 277	1 ⊢ ((B ∈ A ∧ C ∈ A) → (B(I ↾ A)C ↔ B = C))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ 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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