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Theorem brpprod3a 29689
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 29687 . . . . . . 7  |- pprod ( R ,  S )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
21brel 4962 . . . . . 6  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  ->  ( <. X ,  Y >.  e.  ( _V  X.  _V )  /\  Z  e.  ( _V  X.  _V ) ) )
32simprd 461 . . . . 5  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  ->  Z  e.  ( _V  X.  _V )
)
4 elvv 4972 . . . . 5  |-  ( Z  e.  ( _V  X.  _V )  <->  E. z E. w  Z  =  <. z ,  w >. )
53, 4sylib 196 . . . 4  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  ->  E. z E. w  Z  =  <. z ,  w >. )
65pm4.71ri 631 . . 3  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  ( E. z E. w  Z  =  <. z ,  w >.  /\ 
<. X ,  Y >.pprod ( R ,  S ) Z ) )
7 19.41vv 1780 . . 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 252 . 2  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  E. z E. w
( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) Z ) )
9 breq2 4371 . . . 4  |-  ( Z  =  <. z ,  w >.  ->  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. ) )
109pm5.32i 635 . . 3  |-  ( ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) Z )  <-> 
( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <.
z ,  w >. ) )
11102exbii 1676 . 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 3037 . . . . . 6  |-  z  e. 
_V
15 vex 3037 . . . . . 6  |-  w  e. 
_V
1612, 13, 14, 15brpprod 29688 . . . . 5  |-  ( <. X ,  Y >.pprod ( R ,  S )
<. z ,  w >.  <->  ( X R z  /\  Y S w ) )
1716anbi2i 692 . . . 4  |-  ( ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. )  <->  ( Z  =  <. z ,  w >.  /\  ( X R z  /\  Y S w ) ) )
18 3anass 975 . . . 4  |-  ( ( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w )  <->  ( Z  =  <. z ,  w >.  /\  ( X R z  /\  Y S w ) ) )
1917, 18bitr4i 252 . . 3  |-  ( ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. )  <->  ( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w ) )
20192exbii 1676 . 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 271 1  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  E. z E. w
( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399   E.wex 1620    e. wcel 1826   _Vcvv 3034   <.cop 3950   class class class wbr 4367    X. cxp 4911  pprodcpprod 29633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502  df-fv 5504  df-1st 6699  df-2nd 6700  df-txp 29656  df-pprod 29657
This theorem is referenced by:  brpprod3b  29690  brapply  29741  dfrdg4  29753
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