Theorem relopabiv 4903

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopabi 4905. (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 2824	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 2824	. . . . . 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 4420	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6		relopabiv.1	. . 3 ⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
7		df-xp 4780	. . 3 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
8	5, 6, 7	3sstr4i 3289	. 2 ⊢ 𝐴 ⊆ (V × V)
9		df-rel 4781	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
10	8, 9	mpbir 146	1 ⊢ Rel 𝐴

This proof depends on syntax axioms:   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  {copab 4191   × cxp 4772  Rel wrel 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-opab 4193  df-xp 4780  df-rel 4781

This theorem is used by:  relopabv  4904  fvopab4ndm  5803  relfsupp  7287  lgsquadlem1  16208  lgsquadlem2  16209  relsubgr  16508