Theorem opelresi 5016

Description: ⟨𝐴, 𝐴⟩ belongs to a restriction of the identity class iff 𝐴 belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)

Assertion

Ref	Expression
opelresi	⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵))

Proof of Theorem opelresi

Step	Hyp	Ref	Expression
1		opelresg 5012	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴 ∈ 𝐵)))
2		ididg 4875	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴)
3		df-br 4084	. . . 4 ⊢ (𝐴 I 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ I )
4	2, 3	sylib 122	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ ∈ I )
5	4	biantrurd 305	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴 ∈ 𝐵)))
6	1, 5	bitr4d 191	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2200  ⟨cop 3669   class class class wbr 4083   I cid 4379   ↾ cres 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-res 4731

This theorem is referenced by:  issref  5111