Theorem opelidres 5990

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
opelidres	⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵))

Proof of Theorem opelidres

Step	Hyp	Ref	Expression
1		ididg 5839	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴)
2		df-br 5109	. . 3 ⊢ (𝐴 I 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ I )
3	1, 2	sylib 221	. 2 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ ∈ I )
4		opelres 5984	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (𝐴 ∈ 𝐵 ∧ ⟨𝐴, 𝐴⟩ ∈ I )))
5	3, 4	mpbiran2d 720	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2141  ⟨cop 4594   class class class wbr 5108   I cid 5555   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-res 5673

This theorem is referenced by:  dfpo2  6297  ustfilxp  24349  ustelimasn  24359  metustfbas  24693