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Theorem dmxpm 4727
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
dmxpm  |-  ( E. x  x  e.  B  ->  dom  ( A  X.  B )  =  A )
Distinct variable group:    x, B
Allowed substitution hint:    A( x)

Proof of Theorem dmxpm
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2178 . . 3  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
21cbvexv 1870 . 2  |-  ( E. x  x  e.  B  <->  E. z  z  e.  B
)
3 df-xp 4513 . . . 4  |-  ( A  X.  B )  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
43dmeqi 4708 . . 3  |-  dom  ( A  X.  B )  =  dom  { <. y ,  z >.  |  ( y  e.  A  /\  z  e.  B ) }
5 id 19 . . . . 5  |-  ( E. z  z  e.  B  ->  E. z  z  e.  B )
65ralrimivw 2481 . . . 4  |-  ( E. z  z  e.  B  ->  A. y  e.  A  E. z  z  e.  B )
7 dmopab3 4720 . . . 4  |-  ( A. y  e.  A  E. z  z  e.  B  <->  dom 
{ <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }  =  A )
86, 7sylib 121 . . 3  |-  ( E. z  z  e.  B  ->  dom  { <. y ,  z >.  |  ( y  e.  A  /\  z  e.  B ) }  =  A )
94, 8syl5eq 2160 . 2  |-  ( E. z  z  e.  B  ->  dom  ( A  X.  B )  =  A )
102, 9sylbi 120 1  |-  ( E. x  x  e.  B  ->  dom  ( A  X.  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314   E.wex 1451    e. wcel 1463   A.wral 2391   {copab 3956    X. cxp 4505   dom cdm 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-dm 4517
This theorem is referenced by:  rnxpm  4936  ssxpbm  4942  ssxp1  4943  xpexr2m  4948  relrelss  5033  unixpm  5042  exmidfodomrlemim  7021
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