Theorem opabbrex 6593

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 476	. 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉)
2		pm3.41 580	. . . . 5 ⊢ ((𝑥𝑅𝑦 → 𝜑) → ((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
3	2	2alimi 1731	. . . 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 4926	. . 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 4733	1 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583  {copab 4642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-opab 4644

This theorem is referenced by:  fvmptopab1  6594  sprmpt2d  7237  wlkres  26050  opabresex0d  40330  opabresex2d  40334  wlkRes  40858