Theorem opres 5976

Description: Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypothesis

Ref	Expression
opres.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
opres	⊢ (𝐴 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))

Proof of Theorem opres

Step	Hyp	Ref	Expression
1		opres.1	. . 3 ⊢ 𝐵 ∈ V
2	1	opelresi 5974	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (𝐴 ∈ 𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
3	2	baib 543	1 ⊢ (𝐴 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2143  Vcvv 3455  ⟨cop 4589   ↾ cres 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5164  df-xp 5654  df-res 5660

This theorem is referenced by:  resieq  5977  2elresin  6643  mdetunilem9  22681