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Theorem opabbrex 7184
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 487 . 2 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉)
2 pm3.41 495 . . . . 5 ((𝑥𝑅𝑦𝜑) → ((𝑥𝑅𝑦𝜓) → 𝜑))
322alimi 1813 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝜑) → ∀𝑥𝑦((𝑥𝑅𝑦𝜓) → 𝜑))
43adantr 483 . . 3 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → ∀𝑥𝑦((𝑥𝑅𝑦𝜓) → 𝜑))
5 ssopab2 5409 . . 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 5204 1 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wal 1535  wcel 2114  Vcvv 3473  wss 3913   class class class wbr 5042  {copab 5104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-in 3920  df-ss 3930  df-opab 5105
This theorem is referenced by:  opabresex2d  7185  fvmptopab  7186  sprmpod  7868  wlkRes  27418  opabresex0d  43632
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