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Theorem opelresg 4668
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 3592 . . 3 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
21eleq1d 2151 . 2 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷)))
31eleq1d 2151 . . 3 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
43anbi1d 453 . 2 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ ∈ 𝐶𝐴𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
5 vex 2614 . . 3 𝑦 ∈ V
65opelres 4666 . 2 (⟨𝐴, 𝑦⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐶𝐴𝐷))
72, 4, 6vtoclbg 2669 1 (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  cop 3420  cres 4394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-opab 3861  df-xp 4398  df-res 4404
This theorem is referenced by:  brresg  4669  opelresi  4672  issref  4758
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