Theorem opabbrex 7443

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 484	. 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉)
2		pm3.41 492	. . . . 5 ⊢ ((𝑥𝑅𝑦 → 𝜑) → ((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
3	2	2alimi 1812	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
4	3	adantr 480	. . 3 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
5		ssopab2 5509	. . 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 5282	1 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917   class class class wbr 5110  {copab 5172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-in 3924  df-ss 3934  df-opab 5173

This theorem is referenced by:  opabresex2d  7445  fvmptopabOLD  7447  sprmpod  8206  wlkResOLD  29585  opabresex0d  47290