MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opabbrex Structured version   Visualization version   GIF version

Theorem opabbrex 6929
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 478 . 2 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉)
2 pm3.41 487 . . . . 5 ((𝑥𝑅𝑦𝜑) → ((𝑥𝑅𝑦𝜓) → 𝜑))
322alimi 1908 . . . 4 (∀𝑥𝑦(𝑥𝑅𝑦𝜑) → ∀𝑥𝑦((𝑥𝑅𝑦𝜓) → 𝜑))
43adantr 473 . . 3 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → ∀𝑥𝑦((𝑥𝑅𝑦𝜓) → 𝜑))
5 ssopab2 5197 . . 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 5000 1 ((∀𝑥𝑦(𝑥𝑅𝑦𝜑) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651  wcel 2157  Vcvv 3385  wss 3769   class class class wbr 4843  {copab 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783  df-opab 4906
This theorem is referenced by:  opabresex2d  6930  fvmptopab  6931  sprmpt2d  7588  wlkRes  26899  opabresex0d  42140
  Copyright terms: Public domain W3C validator