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Theorem brpprod3b 24498
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 24494 . . 3  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
21breqi 4045 . 2  |-  ( Xpprod ( R ,  S
) <. Y ,  Z >.  <-> 
X `'pprod ( `' R ,  `' S ) <. Y ,  Z >. )
3 brpprod3.1 . . . . 5  |-  X  e. 
_V
4 opex 4253 . . . . 5  |-  <. Y ,  Z >.  e.  _V
53, 4brcnv 4880 . . . 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 24497 . . . 4  |-  ( <. Y ,  Z >.pprod ( `' R ,  `' S
) X  <->  E. z E. w ( X  = 
<. z ,  w >.  /\  Y `' R z  /\  Z `' S w ) )
95, 8bitri 240 . . 3  |-  ( X `'pprod ( `' R ,  `' S ) <. Y ,  Z >. 
<->  E. z E. w
( X  =  <. z ,  w >.  /\  Y `' R z  /\  Z `' S w ) )
10 biid 227 . . . . 5  |-  ( X  =  <. z ,  w >.  <-> 
X  =  <. z ,  w >. )
11 vex 2804 . . . . . 6  |-  z  e. 
_V
126, 11brcnv 4880 . . . . 5  |-  ( Y `' R z  <->  z R Y )
13 vex 2804 . . . . . 6  |-  w  e. 
_V
147, 13brcnv 4880 . . . . 5  |-  ( Z `' S w  <->  w S Z )
1510, 12, 143anbi123i 1140 . . . 4  |-  ( ( X  =  <. z ,  w >.  /\  Y `' R z  /\  Z `' S w )  <->  ( X  =  <. z ,  w >.  /\  z R Y  /\  w S Z ) )
16152exbii 1573 . . 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 240 . 2  |-  ( X `'pprod ( `' R ,  `' S ) <. Y ,  Z >. 
<->  E. z E. w
( X  =  <. z ,  w >.  /\  z R Y  /\  w S Z ) )
182, 17bitri 240 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 176    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039   `'ccnv 4704  pprodcpprod 24445
This theorem is referenced by:  brcart  24542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-txp 24466  df-pprod 24467
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