Theorem bropaex12 5746

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 5104	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		bropaex12.1	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
3	2	eleq2i 2852	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
4	1, 3	bitri 278	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
5		elopaelxp 5745	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
6	4, 5	sylbi 220	. 2 ⊢ (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7		opelxp 5691	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	6, 7	sylib 221	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⟨cop 4590   class class class wbr 5103  {copab 5167   × cxp 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661

This theorem is used by:  fpwwe  10656  efgrelexlema  19877  brslts  28028  rgrprop  30021  rusgrprop  30023  bropabg  44165  clcllaw  49107  asslawass  49109