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Theorem brpprod3b 25732
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:    w, R, z    w, S, z    w, X, z    w, Y, z   
w, Z, z

Proof of Theorem brpprod3b
StepHypRef Expression
1 pprodcnveq 25728 . . 3  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
21breqi 4218 . 2  |-  ( Xpprod ( R ,  S
) <. Y ,  Z >.  <-> 
X `'pprod ( `' R ,  `' S ) <. Y ,  Z >. )
3 brpprod3.1 . . . . 5  |-  X  e. 
_V
4 opex 4427 . . . . 5  |-  <. Y ,  Z >.  e.  _V
53, 4brcnv 5055 . . . 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
86, 7, 3brpprod3a 25731 . . . 4  |-  ( <. Y ,  Z >.pprod ( `' R ,  `' S
) X  <->  E. z E. w ( X  = 
<. z ,  w >.  /\  Y `' R z  /\  Z `' S w ) )
95, 8bitri 241 . . 3  |-  ( X `'pprod ( `' R ,  `' S ) <. Y ,  Z >. 
<->  E. z E. w
( X  =  <. z ,  w >.  /\  Y `' R z  /\  Z `' S w ) )
10 biid 228 . . . . 5  |-  ( X  =  <. z ,  w >.  <-> 
X  =  <. z ,  w >. )
11 vex 2959 . . . . . 6  |-  z  e. 
_V
126, 11brcnv 5055 . . . . 5  |-  ( Y `' R z  <->  z R Y )
13 vex 2959 . . . . . 6  |-  w  e. 
_V
147, 13brcnv 5055 . . . . 5  |-  ( Z `' S w  <->  w S Z )
1510, 12, 143anbi123i 1142 . . . 4  |-  ( ( X  =  <. z ,  w >.  /\  Y `' R z  /\  Z `' S w )  <->  ( X  =  <. z ,  w >.  /\  z R Y  /\  w S Z ) )
16152exbii 1593 . . 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 ) )
179, 16bitri 241 . 2  |-  ( X `'pprod ( `' R ,  `' S ) <. Y ,  Z >. 
<->  E. z E. w
( X  =  <. z ,  w >.  /\  z R Y  /\  w S Z ) )
182, 17bitri 241 1  |-  ( Xpprod ( R ,  S
) <. Y ,  Z >.  <->  E. z E. w ( X  =  <. z ,  w >.  /\  z R Y  /\  w S Z ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   class class class wbr 4212   `'ccnv 4877  pprodcpprod 25675
This theorem is referenced by:  brcart  25777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-txp 25698  df-pprod 25699
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