Theorem opabssi 33005

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 5176	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		eleq1 2853	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
3	2	biimprd 251	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑧 ∈ 𝐴))
4		opabssi.1	. . . . 5 ⊢ (𝜑 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5	3, 4	impel 515	. . . 4 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ 𝐴)
6	5	exlimivv 1965	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ 𝐴)
7	6	abssi 4023	. 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ 𝐴
8	1, 7	eqsstri 3984	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ 𝐴

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743   ⊆ wss 3906  ⟨cop 4597  {copab 5175

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ss 3923  df-opab 5176

This theorem is used by:  opabid2ss  33006