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Theorem brpprod3b 25649
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 25645 . . 3  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
21breqi 4186 . 2  |-  ( Xpprod ( R ,  S
) <. Y ,  Z >.  <-> 
X `'pprod ( `' R ,  `' S ) <. Y ,  Z >. )
3 brpprod3.1 . . . . 5  |-  X  e. 
_V
4 opex 4395 . . . . 5  |-  <. Y ,  Z >.  e.  _V
53, 4brcnv 5022 . . . 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 25648 . . . 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 2927 . . . . . 6  |-  z  e. 
_V
126, 11brcnv 5022 . . . . 5  |-  ( Y `' R z  <->  z R Y )
13 vex 2927 . . . . . 6  |-  w  e. 
_V
147, 13brcnv 5022 . . . . 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 1590 . . 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 1547    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785   class class class wbr 4180   `'ccnv 4844  pprodcpprod 25596
This theorem is referenced by:  brcart  25693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fo 5427  df-fv 5429  df-1st 6316  df-2nd 6317  df-txp 25617  df-pprod 25618
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