Theorem opabbrfex0d 47232

Description: A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.)

Hypotheses

Ref	Expression
opabresex0d.x	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)
opabresex0d.t	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃)
opabresex0d.y	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉)
opabresex0d.c	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

Ref	Expression
opabbrfex0d	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)

Distinct variable groups:   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜃(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabbrfex0d

Step	Hyp	Ref	Expression
1		pm4.24 563	. . 3 ⊢ (𝑥𝑅𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦))
2	1	opabbii 5190	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)}
3		opabresex0d.x	. . 3 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)
4		opabresex0d.t	. . 3 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃)
5		opabresex0d.y	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉)
6		opabresex0d.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
7	3, 4, 5, 6	opabresex0d 47231	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)} ∈ V)
8	2, 7	eqeltrid 2837	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107  {cab 2712  Vcvv 3463   class class class wbr 5123  {copab 5185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-opab 5186  df-xp 5671  df-rel 5672

This theorem is referenced by: (None)