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Theorem dmxp 5063
Description: The domain of a Cartesian 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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4851 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
21dmeqi 5046 . 2  |-  dom  ( A  X.  B )  =  dom  { <. y ,  x >.  |  (
y  e.  A  /\  x  e.  B ) }
3 n0 3651 . . . . 5  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
43biimpi 194 . . . 4  |-  ( B  =/=  (/)  ->  E. x  x  e.  B )
54ralrimivw 2805 . . 3  |-  ( B  =/=  (/)  ->  A. y  e.  A  E. x  x  e.  B )
6 dmopab3 5057 . . 3  |-  ( A. y  e.  A  E. x  x  e.  B  <->  dom 
{ <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }  =  A )
75, 6sylib 196 . 2  |-  ( B  =/=  (/)  ->  dom  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }  =  A )
82, 7syl5eq 2487 1  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2611   A.wral 2720   (/)c0 3642   {copab 4354    X. cxp 4843   dom cdm 4845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-dm 4855
This theorem is referenced by:  dmxpid  5064  rnxp  5273  dmxpss  5274  ssxpb  5277  relrelss  5366  unixp  5375  xpexr2  6524  xpexcnv  6525  frxp  6687  mpt2curryd  6793  fodomr  7467  nqerf  9104  pwsbas  14430  pwsle  14435  imasaddfnlem  14471  imasvscafn  14480  efgrcl  16217  frlmip  18208  txindislem  19211  metustexhalfOLD  20143  metustexhalf  20144  rrxip  20899  dveq0  21477  dv11cn  21478  ismgm  23812  mbfmcst  26679  eulerpartlemt  26759  0rrv  26839  bdayfo  27821  nobndlem3  27840  diophrw  29102
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