Theorem relopabiv 4822

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopabi 4824. (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 2782	. . . . . 6 |- x e. _V
2		vex 2782	. . . . . 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 4345	. . 3 |- { <. x , y >. | ph } C_ { <. x , y >. | ( x e. _V /\ y e. _V ) }
6		relopabiv.1	. . 3 |- A = { <. x , y >. | ph }
7		df-xp 4702	. . 3 |- ( _V X. _V ) = { <. x , y >. | ( x e. _V /\ y e. _V ) }
8	5, 6, 7	3sstr4i 3245	. 2 |- A C_ ( _V X. _V )
9		df-rel 4703	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
10	8, 9	mpbir 146	1 |- Rel A

Syntax hints:   /\ wa 104   = wceq 1375   e. wcel 2180   _Vcvv 2779   C_ wss 3177   {copab 4123   X. cxp 4694   Rel wrel 4701

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-in 3183  df-ss 3190  df-opab 4125  df-xp 4702  df-rel 4703

This theorem is referenced by:  relopabv  4823  lgsquadlem1  15721  lgsquadlem2  15722