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Theorem brpprod3a 24426
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 24424 . . . . . . 7  |- pprod ( R ,  S )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
21brel 4737 . . . . . 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 4748 . . . . 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 1843 . . 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 4027 . . . 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 1570 . 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 2791 . . . . . 6  |-  z  e. 
_V
15 vex 2791 . . . . . 6  |-  w  e. 
_V
1612, 13, 14, 15brpprod 24425 . . . . 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 1570 . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    X. cxp 4687  pprodcpprod 24374
This theorem is referenced by:  brpprod3b  24427  brapply  24477  dfrdg4  24488
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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