MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmxp Unicode version

Theorem dmxp 4885
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 4675 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
21dmeqi 4868 . 2  |-  dom  (  A  X.  B )  =  dom  { <. y ,  x >.  |  (
y  e.  A  /\  x  e.  B ) }
3 n0 3439 . . . . 5  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
43biimpi 188 . . . 4  |-  ( B  =/=  (/)  ->  E. x  x  e.  B )
54ralrimivw 2602 . . 3  |-  ( B  =/=  (/)  ->  A. y  e.  A  E. x  x  e.  B )
6 dmopab3 4879 . . 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 2302 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 2421   A.wral 2518   (/)c0 3430   {copab 4050    X. cxp 4659   dom cdm 4661
This theorem is referenced by:  dmxpid  4886  rnxp  5094  dmxpss  5095  ssxpb  5098  xpexr2  5103  relrelss  5183  unixp  5192  frxp  6159  fodomr  6980  nqerf  8522  pwsbas  13348  pwsle  13353  imasaddfnlem  13392  imasvscafn  13401  efgrcl  14986  txindislem  17289  dveq0  19309  dv11cn  19310  ismgm  20947  axbday  23697  axfelem3  23717  prjcp1  24450  cur1vald  24566  valcurfn1  24571  rngmgmbs3  24784  diophrw  26205  xpexcnv  27027
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-dm 4679
  Copyright terms: Public domain W3C validator