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Theorem opelresg 4796
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 3676 . . 3 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
21eleq1d 2186 . 2 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷)))
31eleq1d 2186 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
43anbi1d 460 . 2 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ ∈ 𝐶𝐴𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
5 vex 2663 . . 3 𝑦 ∈ V
65opelres 4794 . 2 (⟨𝐴, 𝑦⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐶𝐴𝐷))
72, 4, 6vtoclbg 2721 1 (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  cop 3500  cres 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-xp 4515  df-res 4521
This theorem is referenced by:  brresg  4797  opelresi  4800  issref  4891
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