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Theorem brpprod 25730
 Description: Characterize a quatary relationship over a tail cross product. Together with pprodss4v 25729, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brpprod.1
brpprod.2
brpprod.3
brpprod.4
Assertion
Ref Expression
brpprod pprod

Proof of Theorem brpprod
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pprod 25699 . . 3 pprod
21breqi 4218 . 2 pprod
3 opex 4427 . . 3
4 brpprod.3 . . 3
5 brpprod.4 . . 3
63, 4, 5brtxp 25725 . 2
73, 4brco 5043 . . . 4
8 brpprod.1 . . . . . . . . 9
9 brpprod.2 . . . . . . . . 9
108, 9opelvv 4924 . . . . . . . 8
11 vex 2959 . . . . . . . . 9
1211brres 5152 . . . . . . . 8
1310, 12mpbiran2 886 . . . . . . 7
148, 9, 11br1steq 25398 . . . . . . 7
1513, 14bitri 241 . . . . . 6
1615anbi1i 677 . . . . 5
1716exbii 1592 . . . 4
18 breq1 4215 . . . . 5
198, 18ceqsexv 2991 . . . 4
207, 17, 193bitri 263 . . 3
213, 5brco 5043 . . . 4
22 vex 2959 . . . . . . . . 9
2322brres 5152 . . . . . . . 8
2410, 23mpbiran2 886 . . . . . . 7
258, 9, 22br2ndeq 25399 . . . . . . 7
2624, 25bitri 241 . . . . . 6
2726anbi1i 677 . . . . 5
2827exbii 1592 . . . 4
29 breq1 4215 . . . . 5
309, 29ceqsexv 2991 . . . 4
3121, 28, 303bitri 263 . . 3
3220, 31anbi12i 679 . 2
332, 6, 323bitri 263 1 pprod
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725  cvv 2956  cop 3817   class class class wbr 4212   cxp 4876   cres 4880   ccom 4882  c1st 6347  c2nd 6348   ctxp 25674  pprodcpprod 25675 This theorem is referenced by:  brpprod3a  25731 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-txp 25698  df-pprod 25699
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