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Theorem bj-opabssvv 35321
Description: A variant of relopabiv 5730 (which could be proved from it, similarly to relxp 5607 from xpss 5605). (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
StepHypRef Expression
1 vex 3436 . . . . 5 𝑥 ∈ V
2 vex 3436 . . . . 5 𝑦 ∈ V
31, 2pm3.2i 471 . . . 4 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . 3 (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 5463 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6 df-xp 5595 . 2 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
75, 6sseqtrri 3958 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wcel 2106  Vcvv 3432  wss 3887  {copab 5136   × cxp 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-opab 5137  df-xp 5595
This theorem is referenced by: (None)
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