Theorem relopabiv 4786

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopabi 4788. (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 2763	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 2763	. . . . . 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 4309	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		relopabiv.1	. . 3 ⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
7		df-xp 4666	. . 3 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
8	5, 6, 7	3sstr4i 3221	. 2 ⊢ 𝐴 ⊆ (V × V)
9		df-rel 4667	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
10	8, 9	mpbir 146	1 ⊢ Rel 𝐴

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3154  {copab 4090   × cxp 4658  Rel wrel 4665

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167  df-opab 4092  df-xp 4666  df-rel 4667

This theorem is referenced by:  relopabv  4787  lgsquadlem1  15234  lgsquadlem2  15235