Theorem bj-opabssvv 34445

Description: A variant of relopabiv 5693 (which could be proved from it, similarly to relxp 5573 from xpss 5571). (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 3497	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3497	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	pm3.2i 473	. . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 5437	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		df-xp 5561	. 2 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7	5, 6	sseqtrri 4004	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)

Syntax hints:   ∧ wa 398   ∈ wcel 2114  Vcvv 3494   ⊆ wss 3936  {copab 5128   × cxp 5553

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 2793

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3943  df-ss 3952  df-opab 5129  df-xp 5561

This theorem is referenced by: (None)