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Theorem opelres 5831
Description: Ordered pair elementhood in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) (Revised by BJ, 18-Feb-2022.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.)
Assertion
Ref Expression
opelres (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))

Proof of Theorem opelres
StepHypRef Expression
1 df-res 5537 . . 3 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
21elin2 4087 . 2 (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)))
3 opelxp 5561 . . . 4 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ (𝐵𝐴𝐶 ∈ V))
4 elex 3416 . . . . 5 (𝐶𝑉𝐶 ∈ V)
54biantrud 535 . . . 4 (𝐶𝑉 → (𝐵𝐴 ↔ (𝐵𝐴𝐶 ∈ V)))
63, 5bitr4id 293 . . 3 (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ 𝐵𝐴))
76anbi1cd 637 . 2 (𝐶𝑉 → ((⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
82, 7syl5bb 286 1 (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2114  Vcvv 3398  cop 4522   × cxp 5523  cres 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-opab 5093  df-xp 5531  df-res 5537
This theorem is referenced by:  brres  5832  opelresi  5833  opelidres  5837  h2hlm  28915  setsnidel  44363
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