Theorem opabssi 30366

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 5131	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		eleq1 2902	. . . . . 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 508	. . . 4 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ 𝐴)
6	5	exlimivv 1933	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ 𝐴)
7	6	abssi 4048	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ 𝐴
8	1, 7	eqsstri 4003	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ 𝐴

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801   ⊆ wss 3938  ⟨cop 4575  {copab 5130

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 2795

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-in 3945  df-ss 3954  df-opab 5131

This theorem is referenced by:  opabid2ss  30367