Theorem opelresi 5938

Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)

Hypothesis

Ref	Expression
opelresi.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
opelresi	⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))

Proof of Theorem opelresi

Step	Hyp	Ref	Expression
1		opelresi.1	. 2 ⊢ 𝐶 ∈ V
2		opelres 5936	. 2 ⊢ (𝐶 ∈ V → (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
3	1, 2	ax-mp 5	1 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3436  ⟨cop 4583   ↾ cres 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5155  df-xp 5625  df-res 5631

This theorem is referenced by:  opres  5940  dmres  5963  relssres  5973  iss  5986  restidsing  6004  asymref  6065  ssrnres  6127  cnvresima  6179  ressn  6233  funssres  6526  fcnvres  6701  fvn0ssdmfun  7008  relexpindlem  14970  dprd2dlem1  19922  dprd2da  19923  hausdiag  23530  hauseqlcld  23531  ovoliunlem1  25401  undmrnresiss  43597