Theorem dmxpm 4799

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 2217	. . 3 |- ( x = z -> ( x e. B <-> z e. B ) )
2	1	cbvexv 1895	. 2 |- ( E. x x e. B <-> E. z z e. B )
3		df-xp 4585	. . . 4 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
4	3	dmeqi 4780	. . 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 2528	. . . 4 |- ( E. z z e. B -> A. y e. A E. z z e. B )
7		dmopab3 4792	. . . 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 2199	. 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 1332   E.wex 1469   e. wcel 2125   A.wral 2432   {copab 4020   X. cxp 4577   dom cdm 4579

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-xp 4585  df-dm 4589

This theorem is referenced by:  rnxpm  5008  ssxpbm  5014  ssxp1  5015  xpexr2m  5020  relrelss  5105  unixpm  5114  exmidfodomrlemim  7115