Theorem dmxpm 4824

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 2229	. . 3 |- ( x = z -> ( x e. B <-> z e. B ) )
2	1	cbvexv 1906	. 2 |- ( E. x x e. B <-> E. z z e. B )
3		df-xp 4610	. . . 4 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
4	3	dmeqi 4805	. . 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 2540	. . . 4 |- ( E. z z e. B -> A. y e. A E. z z e. B )
7		dmopab3 4817	. . . 4 |- ( A. y e. A E. z z e. B <-> dom { <. y , z >. | ( y e. A /\ z e. B ) } = A )
8	6, 7	sylib 121	. . 3 |- ( E. z z e. B -> dom { <. y , z >. | ( y e. A /\ z e. B ) } = A )
9	4, 8	syl5eq 2211	. 2 |- ( E. z z e. B -> dom ( A X. B ) = A )
10	2, 9	sylbi 120	1 |- ( E. x x e. B -> dom ( A X. B ) = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   A.wral 2444   {copab 4042   X. cxp 4602   dom cdm 4604

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-dm 4614

This theorem is referenced by:  rnxpm  5033  ssxpbm  5039  ssxp1  5040  xpexr2m  5045  relrelss  5130  unixpm  5139  exmidfodomrlemim  7157