Theorem opelresi 5950

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 5948	. 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 3430  ⟨cop 4574   ↾ cres 5630

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 5232  ax-pr 5374

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149  df-xp 5634  df-res 5640

This theorem is referenced by:  opres  5952  dmres  5975  relssres  5985  iss  5998  restidsing  6016  asymref  6077  ssrnres  6140  cnvresima  6192  ressn  6247  funssres  6540  fcnvres  6715  fvn0ssdmfun  7024  relexpindlem  15022  dprd2dlem1  20015  dprd2da  20016  hausdiag  23626  hauseqlcld  23627  ovoliunlem1  25485  undmrnresiss  44057