Theorem relopabv 5770

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2162 and ax-12 2184, see relopab 5773. (Contributed by SN, 8-Sep-2024.)

Assertion

Ref	Expression
relopabv	⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem relopabv

Step	Hyp	Ref	Expression
1		eqid 2736	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	relopabiv 5769	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  {copab 5160  Rel wrel 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-ss 3918  df-opab 5161  df-xp 5630  df-rel 5631

This theorem is referenced by:  opabid2  5777  inopab  5778  difopab  5779  dfres2  6000  cnvopab  6094  cnvopabOLD  6095  funopab  6527  elopabi  8006  relmpoopab  8036  shftfn  14996  cicer  17730  joindmss  18300  meetdmss  18314  lgsquadlem3  27349  tgjustf  28545  perpln1  28782  perpln2  28783  fpwrelmapffslem  32811  fpwrelmap  32812  relfae  34404  satfrel  35561  xpab  35920  vvdifopab  38454  inxprnres  38487  prtlem12  39123  dicvalrelN  41441  diclspsn  41450  dih1dimatlem  41585  rfovcnvf1od  44241