Theorem bropaex12 5676

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

Step	Hyp	Ref	Expression
1		df-br 5079	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		bropaex12.1	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
3	2	eleq2i 2831	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
4	1, 3	bitri 274	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
5		elopaelxp 5675	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
6	4, 5	sylbi 216	. 2 ⊢ (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7		opelxp 5624	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	6, 7	sylib 217	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ⟨cop 4572   class class class wbr 5078  {copab 5140   × cxp 5586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-xp 5594

This theorem is referenced by:  fpwwe  10386  efgrelexlema  19336  rgrprop  27908  rusgrprop  27910  brsslt  33959  clcllaw  45337  asslawass  45339