Theorem relopabiv 4785

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopabi 4787. (Contributed by BJ, 22-Jul-2023.)

Hypothesis

Ref	Expression
relopabiv.1	|- A = { <. x , y >. | ph }

Assertion

Ref	Expression
relopabiv	|- Rel A

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)

Proof of Theorem relopabiv

Step	Hyp	Ref	Expression
1		vex 2763	. . . . . 6 |- x e. _V
2		vex 2763	. . . . . 6 |- y e. _V
3	1, 2	pm3.2i 272	. . . . 5 |- ( x e. _V /\ y e. _V )
4	3	a1i 9	. . . 4 |- ( ph -> ( x e. _V /\ y e. _V ) )
5	4	ssopab2i 4308	. . 3 |- { <. x , y >. | ph } C_ { <. x , y >. | ( x e. _V /\ y e. _V ) }
6		relopabiv.1	. . 3 |- A = { <. x , y >. | ph }
7		df-xp 4665	. . 3 |- ( _V X. _V ) = { <. x , y >. | ( x e. _V /\ y e. _V ) }
8	5, 6, 7	3sstr4i 3220	. 2 |- A C_ ( _V X. _V )
9		df-rel 4666	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
10	8, 9	mpbir 146	1 |- Rel A

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   {copab 4089   X. cxp 4657   Rel wrel 4664

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-opab 4091  df-xp 4665  df-rel 4666

This theorem is referenced by:  relopabv  4786  lgsquadlem1  15191