Theorem opabbrfexd 47266

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

Hypotheses

Ref	Expression
opabresexd.x	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)
opabresexd.y	⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵)
opabresexd.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈)
opabresexd.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉)
opabresexd.c	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

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

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

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabbrfexd

Step	Hyp	Ref	Expression
1		pm4.24 563	. . 3 ⊢ (𝑥𝑅𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦))
2	1	opabbii 5218	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)}
3		opabresexd.x	. . 3 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶)
4		opabresexd.y	. . 3 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵)
5		opabresexd.a	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈)
6		opabresexd.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉)
7		opabresexd.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
8	3, 4, 5, 6, 7	opabresexd 47265	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)} ∈ V)
9	2, 8	eqeltrid 2845	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3481   class class class wbr 5151  {copab 5213  ⟶wf 6565

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-xp 5699  df-rel 5700  df-cnv 5701  df-dm 5703  df-rn 5704  df-fun 6571  df-fn 6572  df-f 6573

This theorem is referenced by: (None)