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Theorem bropaex12 5777
Description: Two classes related by an ordered-pair class abstraction are sets. (Contributed by AV, 21-Jan-2020.)
Hypothesis
Ref Expression
bropaex12.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
Assertion
Ref Expression
bropaex12 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem bropaex12
StepHypRef Expression
1 df-br 5144 . . . 4 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 bropaex12.1 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
32eleq2i 2833 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
41, 3bitri 275 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
5 elopaelxp 5775 . . 3 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
64, 5sylbi 217 . 2 (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7 opelxp 5721 . 2 (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
86, 7sylib 218 1 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691
This theorem is referenced by:  fpwwe  10686  efgrelexlema  19767  brsslt  27830  rgrprop  29578  rusgrprop  29580  bropabg  43336  clcllaw  48107  asslawass  48109
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