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Theorem opelres 5542
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 5540 . 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 382  wcel 2145  Vcvv 3351  cop 4322  cres 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-opab 4847  df-xp 5255  df-res 5261
This theorem is referenced by:  brresOLD  5545  opelresgOLD  5546  opres  5547  dmres  5560  elres  5576  relssres  5578  iss  5588  restidsing  5599  restidsingOLD  5600  asymref  5653  ssrnres  5713  cnvresima  5767  ressn  5815  funssres  6073  fcnvres  6222  fvn0ssdmfun  6493  resiexg  7249  relexpindlem  14011  dprd2dlem1  18648  dprd2da  18649  hausdiag  21669  hauseqlcld  21670  ovoliunlem1  23490  h2hlm  28177  undmrnresiss  38436
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