Theorem bj-opabssvv 37464

Description: A variant of relopabiv 5776 (which could be proved from it, similarly to relxp 5649 from xpss 5647). (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 3433	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3433	. . . . 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 5505	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		df-xp 5637	. 2 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7	5, 6	sseqtrri 3971	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)

Syntax hints:   ∧ wa 395   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889  {copab 5147   × cxp 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-opab 5148  df-xp 5637

This theorem is referenced by: (None)