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Theorem opelresg 5324
 Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
opelresg (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))

Proof of Theorem opelresg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 opeq2 4341 . . 3 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
21eleq1d 2672 . 2 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷)))
31eleq1d 2672 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
43anbi1d 737 . 2 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ ∈ 𝐶𝐴𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
5 vex 3176 . . 3 𝑦 ∈ V
65opelres 5322 . 2 (⟨𝐴, 𝑦⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐶𝐴𝐷))
72, 4, 6vtoclbg 3240 1 (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-res 5050 This theorem is referenced by:  brresg  5325  opelresi  5328  issref  5428
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