Theorem bropaex12 5738

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 5101	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		bropaex12.1	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
3	2	eleq2i 2854	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
4	1, 3	bitri 277	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
5		elopaelxp 5737	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
6	4, 5	sylbi 219	. 2 ⊢ (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7		opelxp 5683	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	6, 7	sylib 220	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ⟨cop 4588   class class class wbr 5100  {copab 5162   × cxp 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653

This theorem is referenced by:  fpwwe  10604  efgrelexlema  19789  brslts  27855  rgrprop  29761  rusgrprop  29763  bropabg  43900  clcllaw  48813  asslawass  48815