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Theorem opelres 5948
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 5650 . . 3 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
21elin2 4162 . 2 (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)))
3 opelxp 5674 . . . 4 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ (𝐵𝐴𝐶 ∈ V))
4 elex 3466 . . . . 5 (𝐶𝑉𝐶 ∈ V)
54biantrud 533 . . . 4 (𝐶𝑉 → (𝐵𝐴 ↔ (𝐵𝐴𝐶 ∈ V)))
63, 5bitr4id 290 . . 3 (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ 𝐵𝐴))
76anbi1cd 635 . 2 (𝐶𝑉 → ((⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
82, 7bitrid 283 1 (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  Vcvv 3448  cop 4597   × cxp 5636  cres 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-xp 5644  df-res 5650
This theorem is referenced by:  brres  5949  opelresi  5950  opelidres  5954  h2hlm  29964  setsnidel  45643
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