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Theorem opelres 5981
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 5681 . . 3 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
21elin2 4192 . 2 (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (⟨𝐵, 𝐶⟩ ∈ 𝑅 ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × V)))
3 opelxp 5705 . . . 4 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ (𝐵𝐴𝐶 ∈ V))
4 elex 3487 . . . . 5 (𝐶𝑉𝐶 ∈ V)
54biantrud 531 . . . 4 (𝐶𝑉 → (𝐵𝐴 ↔ (𝐵𝐴𝐶 ∈ V)))
63, 5bitr4id 290 . . 3 (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝐴 × V) ↔ 𝐵𝐴))
76anbi1cd 633 . 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 395  wcel 2098  Vcvv 3468  cop 4629   × cxp 5667  cres 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-xp 5675  df-res 5681
This theorem is referenced by:  brres  5982  opelresi  5983  opelidres  5987  h2hlm  30742  setsnidel  46614
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