Theorem brxrn2

Description: A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020)

	Ref	Expression
Assertion	brxrn2	|- ( A e. V -> ( A ( R |X. S ) B <-> E. x E. y ( B = <. x , y >. /\ A R x /\ A S y ) ) )

Proof

Step	Hyp	Ref	Expression
1		xrnss3v	|- ( R |X. S ) C_ ( _V X. ( _V X. _V ) )
2	1	brel	|- ( A ( R |X. S ) B -> ( A e. _V /\ B e. ( _V X. _V ) ) )
3	2	simprd	|- ( A ( R |X. S ) B -> B e. ( _V X. _V ) )
4		elvv	|- ( B e. ( _V X. _V ) <-> E. x E. y B = <. x , y >. )
5	3 4	sylib	|- ( A ( R |X. S ) B -> E. x E. y B = <. x , y >. )
6	5	pm4.71ri	|- ( A ( R |X. S ) B <-> ( E. x E. y B = <. x , y >. /\ A ( R |X. S ) B ) )
7		19.41vv	|- ( E. x E. y ( B = <. x , y >. /\ A ( R |X. S ) B ) <-> ( E. x E. y B = <. x , y >. /\ A ( R |X. S ) B ) )
8		breq2	|- ( B = <. x , y >. -> ( A ( R |X. S ) B <-> A ( R |X. S ) <. x , y >. ) )
9	8	pm5.32i	|- ( ( B = <. x , y >. /\ A ( R |X. S ) B ) <-> ( B = <. x , y >. /\ A ( R |X. S ) <. x , y >. ) )
10	9	2exbii	|- ( E. x E. y ( B = <. x , y >. /\ A ( R |X. S ) B ) <-> E. x E. y ( B = <. x , y >. /\ A ( R |X. S ) <. x , y >. ) )
11	6 7 10	3bitr2i	|- ( A ( R |X. S ) B <-> E. x E. y ( B = <. x , y >. /\ A ( R |X. S ) <. x , y >. ) )
12		brxrn	|- ( ( A e. V /\ x e. _V /\ y e. _V ) -> ( A ( R |X. S ) <. x , y >. <-> ( A R x /\ A S y ) ) )
13	12	el3v23	|- ( A e. V -> ( A ( R |X. S ) <. x , y >. <-> ( A R x /\ A S y ) ) )
14	13	anbi2d	|- ( A e. V -> ( ( B = <. x , y >. /\ A ( R |X. S ) <. x , y >. ) <-> ( B = <. x , y >. /\ ( A R x /\ A S y ) ) ) )
15		3anass	|- ( ( B = <. x , y >. /\ A R x /\ A S y ) <-> ( B = <. x , y >. /\ ( A R x /\ A S y ) ) )
16	14 15	bitr4di	|- ( A e. V -> ( ( B = <. x , y >. /\ A ( R |X. S ) <. x , y >. ) <-> ( B = <. x , y >. /\ A R x /\ A S y ) ) )
17	16	2exbidv	|- ( A e. V -> ( E. x E. y ( B = <. x , y >. /\ A ( R |X. S ) <. x , y >. ) <-> E. x E. y ( B = <. x , y >. /\ A R x /\ A S y ) ) )
18	11 17	syl5bb	|- ( A e. V -> ( A ( R |X. S ) B <-> E. x E. y ( B = <. x , y >. /\ A R x /\ A S y ) ) )