Theorem dmxpss 4753

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 2560	. . . 4 |- x e. _V
2	1	eldm2 4533	. . 3 |- ( x e. dom ( A X. B ) <-> E. y <. x , y >. e. ( A X. B ) )
3		opelxp1 4377	. . . 4 |- ( <. x , y >. e. ( A X. B ) -> x e. A )
4	3	exlimiv 1489	. . 3 |- ( E. y <. x , y >. e. ( A X. B ) -> x e. A )
5	2, 4	sylbi 114	. 2 |- ( x e. dom ( A X. B ) -> x e. A )
6	5	ssriv 2949	1 |- dom ( A X. B ) C_ A

Syntax hints:   E.wex 1381   e. wcel 1393   C_ wss 2917   <.cop 3378   X. cxp 4343   dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-dm 4355

This theorem is referenced by:  rnxpss  4754  ssxpbm  4756  ssxp1  4757  funssxp  5060  tfrlemibfn  5942  frecuzrdgfn  9198