Theorem relopabiv 4898

Description: A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopabi 4900. (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 2824	. . . . . 6 |- x e. _V
2		vex 2824	. . . . . 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 4415	. . 3 |- { <. x , y >. | ph } C_ { <. x , y >. | ( x e. _V /\ y e. _V ) }
6		relopabiv.1	. . 3 |- A = { <. x , y >. | ph }
7		df-xp 4775	. . 3 |- ( _V X. _V ) = { <. x , y >. | ( x e. _V /\ y e. _V ) }
8	5, 6, 7	3sstr4i 3289	. 2 |- A C_ ( _V X. _V )
9		df-rel 4776	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
10	8, 9	mpbir 146	1 |- Rel A

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {copab 4186   X. cxp 4767   Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-opab 4188  df-xp 4775  df-rel 4776

This theorem is referenced by:  relopabv  4899  relfsupp  7277  lgsquadlem1  16110  lgsquadlem2  16111  relsubgr  16410