Theorem bj-opabssvv 37355

Description: A variant of relopabiv 5769 (which could be proved from it, similarly to relxp 5642 from xpss 5640). (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 3444	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3444	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	pm3.2i 470	. . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 5498	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		df-xp 5630	. 2 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7	5, 6	sseqtrri 3983	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)

Syntax hints:   ∧ wa 395   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  {copab 5160   × cxp 5622

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-ss 3918  df-opab 5161  df-xp 5630

This theorem is referenced by: (None)