Theorem dmxpss 5051

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 2738	. . . 4 |- x e. _V
2	1	eldm2 4818	. . 3 |- ( x e. dom ( A X. B ) <-> E. y <. x , y >. e. ( A X. B ) )
3		opelxp1 4654	. . . 4 |- ( <. x , y >. e. ( A X. B ) -> x e. A )
4	3	exlimiv 1596	. . 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 3157	1 |- dom ( A X. B ) C_ A

Syntax hints:   E.wex 1490   e. wcel 2146   C_ wss 3127   <.cop 3592   X. cxp 4618   dom cdm 4620

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-dm 4630

This theorem is referenced by:  rnxpss  5052  dmxpss2  5053  ssxpbm  5056  ssxp1  5057  funssxp  5377  tfrlemibfn  6319  tfr1onlembfn  6335  tfrcllembfn  6348  frecuzrdgtcl  10380  frecuzrdgdomlem  10385  dvbssntrcntop  13722