Theorem opabbrex 6862

Description: A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by BJ/AV, 20-Jun-2019.) (Proof shortened by OpenAI, 25-Mar-2020.)

Assertion

Ref	Expression
opabbrex	⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Proof of Theorem opabbrex

Step	Hyp	Ref	Expression
1		simpr 479	. 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉)
2		pm3.41 583	. . . . 5 ⊢ ((𝑥𝑅𝑦 → 𝜑) → ((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
3	2	2alimi 1889	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
4	3	adantr 472	. . 3 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
5		ssopab2 5152	. . 3 ⊢ (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6	4, 5	syl 17	. 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
7	1, 6	ssexd 4958	1 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1630   ∈ wcel 2140  Vcvv 3341   ⊆ wss 3716   class class class wbr 4805  {copab 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-in 3723  df-ss 3730  df-opab 4866

This theorem is referenced by:  opabresex2d  6863  fvmptopab  6864  sprmpt2d  7521  wlkRes  26778  opabresex0d  41831