Metamath Proof Explorer

Theorem dfxrn2

Description: Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022)

Ref Expression
Assertion dfxrn2 
|- ( R |X. S ) = `' { <. <. x , y >. , u >. | ( u R x /\ u S y ) }

Proof

Step Hyp Ref Expression
1 xrnrel 
 |-  Rel ( R |X. S )
2 dfrel4v 
 |-  ( Rel ( R |X. S ) <-> ( R |X. S ) = { <. u , z >. | u ( R |X. S ) z } )
3 1 2 mpbi 
 |-  ( R |X. S ) = { <. u , z >. | u ( R |X. S ) z }
4 breq2 
 |-  ( z = <. x , y >. -> ( u ( R |X. S ) z <-> u ( R |X. S ) <. x , y >. ) )
5 brxrn2 
 |-  ( u e. _V -> ( u ( R |X. S ) z <-> E. x E. y ( z = <. x , y >. /\ u R x /\ u S y ) ) )
6 5 elv 
 |-  ( u ( R |X. S ) z <-> E. x E. y ( z = <. x , y >. /\ u R x /\ u S y ) )
7 brxrn 
 |-  ( ( u e. _V /\ x e. _V /\ y e. _V ) -> ( u ( R |X. S ) <. x , y >. <-> ( u R x /\ u S y ) ) )
8 7 el3v 
 |-  ( u ( R |X. S ) <. x , y >. <-> ( u R x /\ u S y ) )
9 8 anbi2i 
 |-  ( ( z = <. x , y >. /\ u ( R |X. S ) <. x , y >. ) <-> ( z = <. x , y >. /\ ( u R x /\ u S y ) ) )
10 3anass 
 |-  ( ( z = <. x , y >. /\ u R x /\ u S y ) <-> ( z = <. x , y >. /\ ( u R x /\ u S y ) ) )
11 9 10 bitr4i 
 |-  ( ( z = <. x , y >. /\ u ( R |X. S ) <. x , y >. ) <-> ( z = <. x , y >. /\ u R x /\ u S y ) )
12 11 2exbii 
 |-  ( E. x E. y ( z = <. x , y >. /\ u ( R |X. S ) <. x , y >. ) <-> E. x E. y ( z = <. x , y >. /\ u R x /\ u S y ) )
13 4 copsex2gb 
 |-  ( E. x E. y ( z = <. x , y >. /\ u ( R |X. S ) <. x , y >. ) <-> ( z e. ( _V X. _V ) /\ u ( R |X. S ) z ) )
14 6 12 13 3bitr2i 
 |-  ( u ( R |X. S ) z <-> ( z e. ( _V X. _V ) /\ u ( R |X. S ) z ) )
15 14 simplbi 
 |-  ( u ( R |X. S ) z -> z e. ( _V X. _V ) )
16 4 15 cnvoprab 
 |-  `' { <. <. x , y >. , u >. | u ( R |X. S ) <. x , y >. } = { <. u , z >. | u ( R |X. S ) z }
17 8 oprabbii 
 |-  { <. <. x , y >. , u >. | u ( R |X. S ) <. x , y >. } = { <. <. x , y >. , u >. | ( u R x /\ u S y ) }
18 17 cnveqi 
 |-  `' { <. <. x , y >. , u >. | u ( R |X. S ) <. x , y >. } = `' { <. <. x , y >. , u >. | ( u R x /\ u S y ) }
19 3 16 18 3eqtr2i 
 |-  ( R |X. S ) = `' { <. <. x , y >. , u >. | ( u R x /\ u S y ) }