Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brpprod3a Unicode version

Theorem brpprod3a 24497
Description: Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brpprod3.1  |-  X  e. 
_V
brpprod3.2  |-  Y  e. 
_V
brpprod3.3  |-  Z  e. 
_V
Assertion
Ref Expression
brpprod3a  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  E. z E. w
( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w ) )
Distinct variable groups:    w, R, z    w, S, z    w, X, z    w, Y, z   
w, Z, z

Proof of Theorem brpprod3a
StepHypRef Expression
1 pprodss4v 24495 . . . . . . 7  |- pprod ( R ,  S )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
21brel 4753 . . . . . 6  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  ->  ( <. X ,  Y >.  e.  ( _V  X.  _V )  /\  Z  e.  ( _V  X.  _V ) ) )
32simprd 449 . . . . 5  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  ->  Z  e.  ( _V  X.  _V )
)
4 elvv 4764 . . . . 5  |-  ( Z  e.  ( _V  X.  _V )  <->  E. z E. w  Z  =  <. z ,  w >. )
53, 4sylib 188 . . . 4  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  ->  E. z E. w  Z  =  <. z ,  w >. )
65pm4.71ri 614 . . 3  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  ( E. z E. w  Z  =  <. z ,  w >.  /\ 
<. X ,  Y >.pprod ( R ,  S ) Z ) )
7 19.41vv 1855 . . 3  |-  ( E. z E. w ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) Z )  <-> 
( E. z E. w  Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) Z ) )
86, 7bitr4i 243 . 2  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  E. z E. w
( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) Z ) )
9 breq2 4043 . . . 4  |-  ( Z  =  <. z ,  w >.  ->  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. ) )
109pm5.32i 618 . . 3  |-  ( ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) Z )  <-> 
( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <.
z ,  w >. ) )
11102exbii 1573 . 2  |-  ( E. z E. w ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) Z )  <->  E. z E. w ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. ) )
12 brpprod3.1 . . . . . 6  |-  X  e. 
_V
13 brpprod3.2 . . . . . 6  |-  Y  e. 
_V
14 vex 2804 . . . . . 6  |-  z  e. 
_V
15 vex 2804 . . . . . 6  |-  w  e. 
_V
1612, 13, 14, 15brpprod 24496 . . . . 5  |-  ( <. X ,  Y >.pprod ( R ,  S )
<. z ,  w >.  <->  ( X R z  /\  Y S w ) )
1716anbi2i 675 . . . 4  |-  ( ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. )  <->  ( Z  =  <. z ,  w >.  /\  ( X R z  /\  Y S w ) ) )
18 3anass 938 . . . 4  |-  ( ( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w )  <->  ( Z  =  <. z ,  w >.  /\  ( X R z  /\  Y S w ) ) )
1917, 18bitr4i 243 . . 3  |-  ( ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. )  <->  ( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w ) )
20192exbii 1573 . 2  |-  ( E. z E. w ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. )  <->  E. z E. w ( Z  = 
<. z ,  w >.  /\  X R z  /\  Y S w ) )
218, 11, 203bitri 262 1  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  E. z E. w
( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    X. cxp 4703  pprodcpprod 24445
This theorem is referenced by:  brpprod3b  24498  brapply  24548  dfrdg4  24560
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
  Copyright terms: Public domain W3C validator