Theorem opabssi 32776

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.

Step	Hyp	Ref	Expression
1		df-opab 5160	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		eleq1 2849	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
3	2	biimprd 250	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑧 ∈ 𝐴))
4		opabssi.1	. . . . 5 ⊢ (𝜑 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5	3, 4	impel 513	. . . 4 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ 𝐴)
6	5	exlimivv 1951	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ 𝐴)
7	6	abssi 4019	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ 𝐴
8	1, 7	eqsstri 3980	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ 𝐴

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739   ⊆ wss 3902  ⟨cop 4585  {copab 5159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ss 3919  df-opab 5160

This theorem is referenced by:  opabid2ss  32777