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Theorem opelres 5436
 Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
Hypothesis
Ref Expression
opelres.1 𝐵 ∈ V
Assertion
Ref Expression
opelres (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))

Proof of Theorem opelres
StepHypRef Expression
1 opelres.1 . 2 𝐵 ∈ V
2 opelresg 5434 . 2 (𝐵 ∈ V → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
31, 2ax-mp 5 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   ↾ cres 5145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-res 5155 This theorem is referenced by:  brresOLD  5439  opelresgOLD  5440  opres  5441  dmres  5454  elres  5470  relssres  5472  iss  5482  restidsing  5493  restidsingOLD  5494  asymref  5547  ssrnres  5607  cnvresima  5661  ressn  5709  funssres  5968  fcnvres  6120  fvn0ssdmfun  6390  resiexg  7144  relexpindlem  13847  dprd2dlem1  18486  dprd2da  18487  hausdiag  21496  hauseqlcld  21497  ovoliunlem1  23316  h2hlm  27965  undmrnresiss  38227
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