Theorem opabbrex 7501

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 1810	. . . 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 5565	. . 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 5342	1 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976   class class class wbr 5166  {copab 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-opab 5229

This theorem is referenced by:  opabresex2d  7503  fvmptopabOLD  7505  sprmpod  8265  wlkResOLD  29686  opabresex0d  47200