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Theorem relopabiv 5661
Description: A class of ordered pairs is a relation. For a version without disjoint variable condition, but a longer proof using ax-13 2382, see relopabi 5662. (Contributed by BJ, 22-Jul-2023.)
Hypothesis
Ref Expression
relopabiv.1 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
relopabiv Rel 𝐴
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem relopabiv
StepHypRef Expression
1 vex 3447 . . . . . 6 𝑥 ∈ V
2 vex 3447 . . . . . 6 𝑦 ∈ V
31, 2pm3.2i 474 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . . 4 (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 5405 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6 relopabiv.1 . . 3 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
7 df-xp 5529 . . 3 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
85, 6, 73sstr4i 3961 . 2 𝐴 ⊆ (V × V)
9 df-rel 5530 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
108, 9mpbir 234 1 Rel 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  wss 3884  {copab 5095   × cxp 5521  Rel wrel 5528
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-opab 5096  df-xp 5529  df-rel 5530
This theorem is referenced by:  relfldext  31124
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