Theorem opres 5958

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 5956	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (𝐴 ∈ 𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
3	2	baib 535	1 ⊢ (𝐴 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588   ↾ cres 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5640  df-res 5646

This theorem is referenced by:  resieq  5959  2elresin  6623  mdetunilem9  22581