Theorem bj-opabssvv 37794

Description: A variant of relopabiv 5807 (which could be proved from it, similarly to relxp 5679 from xpss 5677). (Contributed by BJ, 28-Dec-2023.)

Assertion

Ref	Expression
bj-opabssvv	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-opabssvv

Step	Hyp	Ref	Expression
1		vex 3459	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	pm3.2i 475	. . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 5535	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		df-xp 5667	. 2 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7	5, 6	sseqtrri 3986	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)

Syntax hints:   ∧ wa 400   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  {copab 5173   × cxp 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3922  df-opab 5174  df-xp 5667

This theorem is referenced by: (None)