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Theorem bropaex12 5780
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 5149 . . . 4 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 bropaex12.1 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
32eleq2i 2831 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
41, 3bitri 275 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
5 elopaelxp 5778 . . 3 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
64, 5sylbi 217 . 2 (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7 opelxp 5725 . 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 1537  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148  {copab 5210   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695
This theorem is referenced by:  fpwwe  10684  efgrelexlema  19782  brsslt  27845  rgrprop  29593  rusgrprop  29595  bropabg  43313  clcllaw  48035  asslawass  48037
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