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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 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)

Proof of Theorem opabbrex
StepHypRef Expression
1 simpr 479 . 2 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉)
2 pm3.41 583 . . . . 5 ((𝑥𝑅𝑦𝜑) → ((𝑥𝑅𝑦𝜓) → 𝜑))
322alimi 1889 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝜑) → ∀𝑥𝑦((𝑥𝑅𝑦𝜓) → 𝜑))
43adantr 472 . . 3 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → ∀𝑥𝑦((𝑥𝑅𝑦𝜓) → 𝜑))
5 ssopab2 5152 . . 3 (∀𝑥𝑦((𝑥𝑅𝑦𝜓) → 𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
64, 5syl 17 . 2 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
71, 6ssexd 4958 1 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Colors of variables: wff setvar class
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
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