Theorem dmxpss 4905

Description: The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)

Assertion

Ref	Expression
dmxpss	|- dom ( A X. B ) C_ A

Proof of Theorem dmxpss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2644	. . . 4 |- x e. _V
2	1	eldm2 4675	. . 3 |- ( x e. dom ( A X. B ) <-> E. y <. x , y >. e. ( A X. B ) )
3		opelxp1 4511	. . . 4 |- ( <. x , y >. e. ( A X. B ) -> x e. A )
4	3	exlimiv 1545	. . 3 |- ( E. y <. x , y >. e. ( A X. B ) -> x e. A )
5	2, 4	sylbi 120	. 2 |- ( x e. dom ( A X. B ) -> x e. A )
6	5	ssriv 3051	1 |- dom ( A X. B ) C_ A

Syntax hints:   E.wex 1436   e. wcel 1448   C_ wss 3021   <.cop 3477   X. cxp 4475   dom cdm 4477

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-dm 4487

This theorem is referenced by:  rnxpss  4906  dmxpss2  4907  ssxpbm  4910  ssxp1  4911  funssxp  5228  tfrlemibfn  6155  tfr1onlembfn  6171  tfrcllembfn  6184  frecuzrdgtcl  10026  frecuzrdgdomlem  10031  dvbssntrcntop  12526