Metamath Proof Explorer


Theorem brpprod3b

Description: Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014)

Ref Expression
Hypotheses brpprod3.1
|- X e. _V
brpprod3.2
|- Y e. _V
brpprod3.3
|- Z e. _V
Assertion brpprod3b
|- ( X pprod ( R , S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) )

Proof

Step Hyp Ref Expression
1 brpprod3.1
 |-  X e. _V
2 brpprod3.2
 |-  Y e. _V
3 brpprod3.3
 |-  Z e. _V
4 pprodcnveq
 |-  pprod ( R , S ) = `' pprod ( `' R , `' S )
5 4 breqi
 |-  ( X pprod ( R , S ) <. Y , Z >. <-> X `' pprod ( `' R , `' S ) <. Y , Z >. )
6 opex
 |-  <. Y , Z >. e. _V
7 1 6 brcnv
 |-  ( X `' pprod ( `' R , `' S ) <. Y , Z >. <-> <. Y , Z >. pprod ( `' R , `' S ) X )
8 2 3 1 brpprod3a
 |-  ( <. Y , Z >. pprod ( `' R , `' S ) X <-> E. z E. w ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) )
9 7 8 bitri
 |-  ( X `' pprod ( `' R , `' S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) )
10 biid
 |-  ( X = <. z , w >. <-> X = <. z , w >. )
11 vex
 |-  z e. _V
12 2 11 brcnv
 |-  ( Y `' R z <-> z R Y )
13 vex
 |-  w e. _V
14 3 13 brcnv
 |-  ( Z `' S w <-> w S Z )
15 10 12 14 3anbi123i
 |-  ( ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) <-> ( X = <. z , w >. /\ z R Y /\ w S Z ) )
16 15 2exbii
 |-  ( E. z E. w ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) )
17 9 16 bitri
 |-  ( X `' pprod ( `' R , `' S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) )
18 5 17 bitri
 |-  ( X pprod ( R , S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) )