Theorem bropaex12 5730

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 5108	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		bropaex12.1	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
3	2	eleq2i 2820	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
4	1, 3	bitri 275	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
5		elopaelxp 5728	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
6	4, 5	sylbi 217	. 2 ⊢ (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7		opelxp 5674	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	6, 7	sylib 218	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   class class class wbr 5107  {copab 5169   × cxp 5636

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644

This theorem is referenced by:  fpwwe  10599  efgrelexlema  19679  brsslt  27697  rgrprop  29488  rusgrprop  29490  bropabg  43312  clcllaw  48179  asslawass  48181