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Theorem brabvv 5608
Description: If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.)
Assertion
Ref Expression
brabvv (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Distinct variable groups:   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem brabvv
StepHypRef Expression
1 df-br 3807 . . . . . 6 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2 elopab 4043 . . . . . 6 (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
31, 2bitri 182 . . . . 5 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 exsimpl 1549 . . . . . 6 (∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
54eximi 1532 . . . . 5 (∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
63, 5sylbi 119 . . . 4 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → ∃𝑥𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
7 vex 2614 . . . . . . . 8 𝑥 ∈ V
8 vex 2614 . . . . . . . 8 𝑦 ∈ V
97, 8opth 4022 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
109biimpi 118 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ → (𝑥 = 𝑋𝑦 = 𝑌))
1110eqcoms 2086 . . . . 5 (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝑋𝑦 = 𝑌))
12112eximi 1533 . . . 4 (∃𝑥𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦(𝑥 = 𝑋𝑦 = 𝑌))
136, 12syl 14 . . 3 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → ∃𝑥𝑦(𝑥 = 𝑋𝑦 = 𝑌))
14 eeanv 1850 . . 3 (∃𝑥𝑦(𝑥 = 𝑋𝑦 = 𝑌) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
1513, 14sylib 120 . 2 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
16 isset 2615 . . 3 (𝑋 ∈ V ↔ ∃𝑥 𝑥 = 𝑋)
17 isset 2615 . . 3 (𝑌 ∈ V ↔ ∃𝑦 𝑦 = 𝑌)
1816, 17anbi12i 448 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
1915, 18sylibr 132 1 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  Vcvv 2611  cop 3420   class class class wbr 3806  {copab 3859
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 3994
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-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861
This theorem is referenced by: (None)
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