Theorem opelresi 5956

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 5954	. 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 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:  opres  5958  dmres  5981  relssres  5991  iss  6004  restidsing  6022  asymref  6083  ssrnres  6146  cnvresima  6198  ressn  6253  funssres  6546  fcnvres  6721  fvn0ssdmfun  7030  relexpindlem  15000  dprd2dlem1  19989  dprd2da  19990  hausdiag  23606  hauseqlcld  23607  ovoliunlem1  25476  undmrnresiss  43989