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Theorem dmxp 4804
Description: The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmxp  |-  ( B  =/=  (/)  ->  dom  (  A  X.  B )  =  A )

Proof of Theorem dmxp
StepHypRef Expression
1 df-xp 4594 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
21dmeqi 4787 . 2  |-  dom  (  A  X.  B )  =  dom  { <. y ,  x >.  |  (
y  e.  A  /\  x  e.  B ) }
3 n0 3371 . . . . 5  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
43biimpi 188 . . . 4  |-  ( B  =/=  (/)  ->  E. x  x  e.  B )
54ralrimivw 2589 . . 3  |-  ( B  =/=  (/)  ->  A. y  e.  A  E. x  x  e.  B )
6 dmopab3 4798 . . 3  |-  ( A. y  e.  A  E. x  x  e.  B  <->  dom 
{ <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }  =  A )
75, 6sylib 190 . 2  |-  ( B  =/=  (/)  ->  dom  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }  =  A )
82, 7syl5eq 2297 1  |-  ( B  =/=  (/)  ->  dom  (  A  X.  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   (/)c0 3362   {copab 3973    X. cxp 4578   dom cdm 4580
This theorem is referenced by:  dmxpid  4805  rnxp  5013  dmxpss  5014  ssxpb  5017  xpexr2  5022  relrelss  5102  unixp  5111  frxp  6077  fodomr  6897  nqerf  8434  pwsbas  13260  pwsle  13265  imasaddfnlem  13304  imasvscafn  13313  efgrcl  14859  txindislem  17159  dveq0  19179  dv11cn  19180  ismgm  20817  axbday  23496  axfelem3  23516  prjcp1  24249  cur1vald  24365  valcurfn1  24370  rngmgmbs3  24583  diophrw  26004  xpexcnv  26826
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-dm 4598
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