Theorem dmxpm 4882

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.

Step	Hyp	Ref	Expression
1		eleq1 2256	. . 3 |- ( x = z -> ( x e. B <-> z e. B ) )
2	1	cbvexv 1930	. 2 |- ( E. x x e. B <-> E. z z e. B )
3		df-xp 4665	. . . 4 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
4	3	dmeqi 4863	. . 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 )
6	5	ralrimivw 2568	. . . 4 |- ( E. z z e. B -> A. y e. A E. z z e. B )
7		dmopab3 4875	. . . 4 |- ( A. y e. A E. z z e. B <-> dom { <. y , z >. | ( y e. A /\ z e. B ) } = A )
8	6, 7	sylib 122	. . 3 |- ( E. z z e. B -> dom { <. y , z >. | ( y e. A /\ z e. B ) } = A )
9	4, 8	eqtrid 2238	. 2 |- ( E. z z e. B -> dom ( A X. B ) = A )
10	2, 9	sylbi 121	1 |- ( E. x x e. B -> dom ( A X. B ) = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   {copab 4089   X. cxp 4657   dom cdm 4659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-dm 4669

This theorem is referenced by:  rnxpm  5095  ssxpbm  5101  ssxp1  5102  xpexr2m  5107  relrelss  5192  unixpm  5201  exmidfodomrlemim  7261  imasaddfnlemg  12897