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Theorem brpprod3b 24427
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 24423 . . 3  |- pprod ( R ,  S )  =  `'pprod ( `' R ,  `' S )
21breqi 4029 . 2  |-  ( Xpprod ( R ,  S
) <. Y ,  Z >.  <-> 
X `'pprod ( `' R ,  `' S ) <. Y ,  Z >. )
3 brpprod3.1 . . . . 5  |-  X  e. 
_V
4 opex 4237 . . . . 5  |-  <. Y ,  Z >.  e.  _V
53, 4brcnv 4864 . . . 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 24426 . . . 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 2791 . . . . . 6  |-  z  e. 
_V
126, 11brcnv 4864 . . . . 5  |-  ( Y `' R z  <->  z R Y )
13 vex 2791 . . . . . 6  |-  w  e. 
_V
147, 13brcnv 4864 . . . . 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 1570 . . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   `'ccnv 4688  pprodcpprod 24374
This theorem is referenced by:  brcart  24471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-txp 24395  df-pprod 24396
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