Theorem opelresi 5945

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 5943	. 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 208   ∧ wa 397   ∈ wcel 2121  Vcvv 3433  ⟨cop 4563   ↾ cres 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5137  df-xp 5626  df-res 5632

This theorem is referenced by:  opres  5947  dmres  5970  relssres  5980  iss  5993  restidsing  6011  asymref  6072  ssrnres  6132  cnvresima  6184  ressn  6239  funssres  6532  fcnvres  6707  fvn0ssdmfun  7018  relexpindlem  15020  dprd2dlem1  20012  dprd2da  20013  hausdiag  23631  hauseqlcld  23632  ovoliunlem1  25490  undmrnresiss  44061