Theorem opelresi 5985

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 5983	. 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))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∈ wcel 2142  Vcvv 3454  ⟨cop 4594   ↾ cres 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173  df-xp 5666  df-res 5672

This theorem is used by:  opres  5987  dmres  6010  relssres  6020  iss  6036  restidsing  6054  asymref  6115  ssrnres  6175  cnvresima  6230  ressn  6286  funssres  6580  fcnvres  6755  fvn0ssdmfun  7069  relexpindlem  15107  dprd2dlem1  20119  dprd2da  20120  hausdiag  23813  hauseqlcld  23814  ovoliunlem1  25672  undmrnresiss  44358