Theorem opelresi 4917

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 4913	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴 ∈ 𝐵)))
2		ididg 4779	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴)
3		df-br 4003	. . . 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 2148  ⟨cop 3595   class class class wbr 4002   I cid 4287   ↾ cres 4627

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-res 4637

This theorem is referenced by:  issref  5010