Theorem relopabv 4790

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopab 4792. (Contributed by SN, 8-Sep-2024.)

Assertion

Ref	Expression
relopabv	|- Rel { <. x , y >. | ph }

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem relopabv

Step	Hyp	Ref	Expression
1		eqid 2196	. 2 |- { <. x , y >. | ph } = { <. x , y >. | ph }
2	1	relopabiv 4789	1 |- Rel { <. x , y >. | ph }

Syntax hints:   {copab 4093   Rel wrel 4668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-opab 4095  df-xp 4669  df-rel 4670

This theorem is referenced by:  lgsquadlem3  15320