Theorem relopabv 5807

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2191 and ax-12 2212, see relopab 5810. (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 2762	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	relopabiv 5806	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

This proof depends on syntax axioms:  {copab 5172  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-ss 3921  df-opab 5173  df-xp 5666  df-rel 5667

This theorem is used by:  opabid2  5814  inopab  5815  difopab  5816  dfres2  6042  cnvopab  6136  funopab  6571  elopabi  8057  relmpoopab  8087  shftfn  15117  cicer  17869  joindmss  18439  meetdmss  18453  lgsquadlem3  27557  tgjustf  28753  perpln1  29001  perpln2  29002  fpwrelmapffslem  33088  fpwrelmap  33089  relfae  34646  satfrel  35867  xpab  36226  vvdifopab  38942  inxprnres  38975  prtlem12  39669  dicvalrelN  41987  diclspsn  41996  dih1dimatlem  42131  rfovcnvf1od  44758