Theorem dmxpss 5165

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 2803	. . . 4 |- x e. _V
2	1	eldm2 4927	. . 3 |- ( x e. dom ( A X. B ) <-> E. y <. x , y >. e. ( A X. B ) )
3		opelxp1 4757	. . . 4 |- ( <. x , y >. e. ( A X. B ) -> x e. A )
4	3	exlimiv 1644	. . 3 |- ( E. y <. x , y >. e. ( A X. B ) -> x e. A )
5	2, 4	sylbi 121	. 2 |- ( x e. dom ( A X. B ) -> x e. A )
6	5	ssriv 3229	1 |- dom ( A X. B ) C_ A

Syntax hints:   E.wex 1538   e. wcel 2200   C_ wss 3198   <.cop 3670   X. cxp 4721   dom cdm 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-dm 4733

This theorem is referenced by:  rnxpss  5166  dmxpss2  5167  ssxpbm  5170  ssxp1  5171  funssxp  5501  tfrlemibfn  6489  tfr1onlembfn  6505  tfrcllembfn  6518  frecuzrdgtcl  10664  frecuzrdgdomlem  10669  dvbssntrcntop  15398