Theorem relopabiv 4877

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopabi 4879. (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

Step	Hyp	Ref	Expression
1		vex 2815	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 2815	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	pm3.2i 272	. . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 9	. . . 4 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 4395	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		relopabiv.1	. . 3 ⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
7		df-xp 4754	. . 3 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
8	5, 6, 7	3sstr4i 3278	. 2 ⊢ 𝐴 ⊆ (V × V)
9		df-rel 4755	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
10	8, 9	mpbir 146	1 ⊢ Rel 𝐴

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ⊆ wss 3210  {copab 4169   × cxp 4746  Rel wrel 4753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-opab 4171  df-xp 4754  df-rel 4755

This theorem is referenced by:  relopabv  4878  relfsupp  7239  lgsquadlem1  15937  lgsquadlem2  15938  relsubgr  16237