Theorem opabbrex 6695

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 477	. 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉)
2		pm3.41 582	. . . . 5 ⊢ ((𝑥𝑅𝑦 → 𝜑) → ((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
3	2	2alimi 1740	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
4	3	adantr 481	. . 3 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝜓) → 𝜑))
5		ssopab2 5001	. . 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 4805	1 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653  {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-opab 4713

This theorem is referenced by:  opabresex2d  6696  fvmptopab  6697  sprmpt2d  7350  wlkRes  26546  opabresex0d  41304