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Theorem brpprod3a 30702
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 30700 . . . . . . 7  |- pprod ( R ,  S )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
21brel 4902 . . . . . 6  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  ->  ( <. X ,  Y >.  e.  ( _V  X.  _V )  /\  Z  e.  ( _V  X.  _V ) ) )
32simprd 469 . . . . 5  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  ->  Z  e.  ( _V  X.  _V )
)
4 elvv 4912 . . . . 5  |-  ( Z  e.  ( _V  X.  _V )  <->  E. z E. w  Z  =  <. z ,  w >. )
53, 4sylib 201 . . . 4  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  ->  E. z E. w  Z  =  <. z ,  w >. )
65pm4.71ri 643 . . 3  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  ( E. z E. w  Z  =  <. z ,  w >.  /\ 
<. X ,  Y >.pprod ( R ,  S ) Z ) )
7 19.41vv 1842 . . 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 260 . 2  |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  E. z E. w
( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) Z ) )
9 breq2 4420 . . . 4  |-  ( Z  =  <. z ,  w >.  ->  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. ) )
109pm5.32i 647 . . 3  |-  ( ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) Z )  <-> 
( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <.
z ,  w >. ) )
11102exbii 1730 . 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 3060 . . . . . 6  |-  z  e. 
_V
15 vex 3060 . . . . . 6  |-  w  e. 
_V
1612, 13, 14, 15brpprod 30701 . . . . 5  |-  ( <. X ,  Y >.pprod ( R ,  S )
<. z ,  w >.  <->  ( X R z  /\  Y S w ) )
1716anbi2i 705 . . . 4  |-  ( ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. )  <->  ( Z  =  <. z ,  w >.  /\  ( X R z  /\  Y S w ) ) )
18 3anass 995 . . . 4  |-  ( ( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w )  <->  ( Z  =  <. z ,  w >.  /\  ( X R z  /\  Y S w ) ) )
1917, 18bitr4i 260 . . 3  |-  ( ( Z  =  <. z ,  w >.  /\  <. X ,  Y >.pprod ( R ,  S ) <. z ,  w >. )  <->  ( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w ) )
20192exbii 1730 . 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 279 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 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898   _Vcvv 3057   <.cop 3986   class class class wbr 4416    X. cxp 4851  pprodcpprod 30646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-1st 6820  df-2nd 6821  df-txp 30669  df-pprod 30670
This theorem is referenced by:  brpprod3b  30703  brapply  30754  dfrdg4  30767
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