Theorem relopabv 5830

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2156 and ax-12 2176, see relopab 5833. (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 5829	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  {copab 5204  Rel wrel 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-opab 5205  df-xp 5690  df-rel 5691

This theorem is referenced by:  opabid2  5837  inopab  5838  difopab  5839  difopabOLD  5840  dfres2  6058  cnvopab  6156  cnvopabOLD  6157  funopab  6600  elopabi  8088  relmpoopab  8120  shftfn  15113  cicer  17851  joindmss  18425  meetdmss  18439  lgsquadlem3  27427  tgjustf  28482  perpln1  28719  perpln2  28720  fpwrelmapffslem  32744  fpwrelmap  32745  relfae  34249  satfrel  35373  xpab  35727  vvdifopab  38262  inxprnres  38294  prtlem12  38869  dicvalrelN  41188  diclspsn  41197  dih1dimatlem  41332  rfovcnvf1od  44022