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Theorem opabssi 30380
 Description: Sufficient condition for a collection of ordered pairs to be a subclass of a relation. (Contributed by Peter Mazsa, 21-Oct-2019.) (Revised by Thierry Arnoux, 18-Feb-2022.)
Hypothesis
Ref Expression
opabssi.1 (𝜑 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
Assertion
Ref Expression
opabssi {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ 𝐴
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabssi
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-opab 5096 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 eleq1 2880 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
32biimprd 251 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑧𝐴))
4 opabssi.1 . . . . 5 (𝜑 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
53, 4impel 509 . . . 4 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧𝐴)
65exlimivv 1933 . . 3 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧𝐴)
76abssi 4000 . 2 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ 𝐴
81, 7eqsstri 3952 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ 𝐴
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  {cab 2779   ⊆ wss 3884  ⟨cop 4534  {copab 5095 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-opab 5096 This theorem is referenced by:  opabid2ss  30381
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