Theorem brpprod3b 31994

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

Hypotheses

Ref	Expression
brpprod3.1	|- X e. _V
brpprod3.2	|- Y e. _V
brpprod3.3	|- Z e. _V

Assertion

Ref	Expression
brpprod3b	|- ( Xpprod ( R , S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) )

Distinct variable groups:   z, w, R   w, S, z   w, X, z   w, Y, z   w, Z, z

Proof of Theorem brpprod3b

Step	Hyp	Ref	Expression
1		pprodcnveq 31990	. . 3 |- pprod ( R , S ) = `'pprod ( `' R , `' S )
2	1	breqi 4659	. 2 |- ( Xpprod ( R , S ) <. Y , Z >. <-> X `'pprod ( `' R , `' S ) <. Y , Z >. )
3		brpprod3.1	. . . . 5 |- X e. _V
4		opex 4932	. . . . 5 |- <. Y , Z >. e. _V
5	3, 4	brcnv 5305	. . . 4 |- ( X `'pprod ( `' R , `' S ) <. Y , Z >. <-> <. Y , Z >.pprod ( `' R , `' S ) X )
6		brpprod3.2	. . . . 5 |- Y e. _V
7		brpprod3.3	. . . . 5 |- Z e. _V
8	6, 7, 3	brpprod3a 31993	. . . 4 |- ( <. Y , Z >.pprod ( `' R , `' S ) X <-> E. z E. w ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) )
9	5, 8	bitri 264	. . 3 |- ( X `'pprod ( `' R , `' S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) )
10		biid 251	. . . . 5 |- ( X = <. z , w >. <-> X = <. z , w >. )
11		vex 3203	. . . . . 6 |- z e. _V
12	6, 11	brcnv 5305	. . . . 5 |- ( Y `' R z <-> z R Y )
13		vex 3203	. . . . . 6 |- w e. _V
14	7, 13	brcnv 5305	. . . . 5 |- ( Z `' S w <-> w S Z )
15	10, 12, 14	3anbi123i 1251	. . . 4 |- ( ( X = <. z , w >. /\ Y `' R z /\ Z `' S w ) <-> ( X = <. z , w >. /\ z R Y /\ w S Z ) )
16	15	2exbii 1775	. . 3 |- ( 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 264	. 2 |- ( X `'pprod ( `' R , `' S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) )
18	2, 17	bitri 264	1 |- ( Xpprod ( R , S ) <. Y , Z >. <-> E. z E. w ( X = <. z , w >. /\ z R Y /\ w S Z ) )

Syntax hints:   <-> wb 196   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   `'ccnv 5113  pprodcpprod 31938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-pprod 31962

This theorem is referenced by:  brcart  32039