Theorem relopabv 5822

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2155 and ax-12 2172, see relopab 5825. (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 2733	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	relopabiv 5821	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  {copab 5211  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684

This theorem is referenced by:  opabid2  5829  inopab  5830  difopab  5831  difopabOLD  5832  dfres2  6042  cnvopab  6139  funopab  6584  elopabi  8048  relmpoopab  8080  shftfn  15020  cicer  17753  joindmss  18332  meetdmss  18346  lgsquadlem3  26885  tgjustf  27724  perpln1  27961  perpln2  27962  fpwrelmapffslem  31957  fpwrelmap  31958  relfae  33245  satfrel  34358  xpab  34695  vvdifopab  37128  inxprnres  37161  prtlem12  37737  dicvalrelN  40056  diclspsn  40065  dih1dimatlem  40200  rfovcnvf1od  42755