Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-opabssvv Structured version   Visualization version   GIF version

Theorem bj-opabssvv 34974
Description: A variant of relopabiv 5674 (which could be proved from it, similarly to relxp 5553 from xpss 5551). (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 3404 . . . . 5 𝑥 ∈ V
2 vex 3404 . . . . 5 𝑦 ∈ V
31, 2pm3.2i 474 . . . 4 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . 3 (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 5415 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6 df-xp 5541 . 2 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
75, 6sseqtrri 3924 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
Colors of variables: wff setvar class
Syntax hints:  wa 399  wcel 2114  Vcvv 3400  wss 3853  {copab 5102   × cxp 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3402  df-in 3860  df-ss 3870  df-opab 5103  df-xp 5541
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator