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Theorem brabvv 5578
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 3792 . . . . . 6 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2 elopab 4022 . . . . . 6 (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
31, 2bitri 177 . . . . 5 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 ↔ ∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 exsimpl 1524 . . . . . 6 (∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
54eximi 1507 . . . . 5 (∃𝑥𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
63, 5sylbi 118 . . . 4 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → ∃𝑥𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)
7 vex 2577 . . . . . . . 8 𝑥 ∈ V
8 vex 2577 . . . . . . . 8 𝑦 ∈ V
97, 8opth 4001 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ ↔ (𝑥 = 𝑋𝑦 = 𝑌))
109biimpi 117 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ → (𝑥 = 𝑋𝑦 = 𝑌))
1110eqcoms 2059 . . . . 5 (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑥 = 𝑋𝑦 = 𝑌))
12112eximi 1508 . . . 4 (∃𝑥𝑦𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑦(𝑥 = 𝑋𝑦 = 𝑌))
136, 12syl 14 . . 3 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → ∃𝑥𝑦(𝑥 = 𝑋𝑦 = 𝑌))
14 eeanv 1823 . . 3 (∃𝑥𝑦(𝑥 = 𝑋𝑦 = 𝑌) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
1513, 14sylib 131 . 2 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
16 isset 2578 . . 3 (𝑋 ∈ V ↔ ∃𝑥 𝑥 = 𝑋)
17 isset 2578 . . 3 (𝑌 ∈ V ↔ ∃𝑦 𝑦 = 𝑌)
1816, 17anbi12i 441 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) ↔ (∃𝑥 𝑥 = 𝑋 ∧ ∃𝑦 𝑦 = 𝑌))
1915, 18sylibr 141 1 (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  cop 3405   class class class wbr 3791  {copab 3844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846
This theorem is referenced by: (None)
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