Theorem dmxpm 4958

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 2294	. . 3 |- ( x = z -> ( x e. B <-> z e. B ) )
2	1	cbvexv 1967	. 2 |- ( E. x x e. B <-> E. z z e. B )
3		df-xp 4737	. . . 4 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
4	3	dmeqi 4938	. . 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 2607	. . . 4 |- ( E. z z e. B -> A. y e. A E. z z e. B )
7		dmopab3 4950	. . . 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 2276	. 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 1398   E.wex 1541   e. wcel 2202   A.wral 2511   {copab 4154   X. cxp 4729   dom cdm 4731

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-dm 4741

This theorem is referenced by:  xpexcnvm  5098  rnxpm  5173  ssxpbm  5179  ssxp1  5180  xpexr2m  5185  relrelss  5270  unixpm  5279  exmidfodomrlemim  7455  pwsbas  13438  imasaddfnlemg  13460