Metamath Proof Explorer


Theorem relssdmrnOLD

Description: Obsolete version of relssdmrn as of 23-Dec-2024. (Contributed by NM, 3-Aug-1994) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion relssdmrnOLD Rel A A dom A × ran A

Proof

Step Hyp Ref Expression
1 id Rel A Rel A
2 19.8a x y A y x y A
3 19.8a x y A x x y A
4 opelxp x y dom A × ran A x dom A y ran A
5 vex x V
6 5 eldm2 x dom A y x y A
7 vex y V
8 7 elrn2 y ran A x x y A
9 6 8 anbi12i x dom A y ran A y x y A x x y A
10 4 9 bitri x y dom A × ran A y x y A x x y A
11 2 3 10 sylanbrc x y A x y dom A × ran A
12 11 a1i Rel A x y A x y dom A × ran A
13 1 12 relssdv Rel A A dom A × ran A