Theorem dmxpss 5132

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 2779	. . . 4 |- x e. _V
2	1	eldm2 4895	. . 3 |- ( x e. dom ( A X. B ) <-> E. y <. x , y >. e. ( A X. B ) )
3		opelxp1 4727	. . . 4 |- ( <. x , y >. e. ( A X. B ) -> x e. A )
4	3	exlimiv 1622	. . 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 3205	1 |- dom ( A X. B ) C_ A

Syntax hints:   E.wex 1516   e. wcel 2178   C_ wss 3174   <.cop 3646   X. cxp 4691   dom cdm 4693

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-dm 4703

This theorem is referenced by:  rnxpss  5133  dmxpss2  5134  ssxpbm  5137  ssxp1  5138  funssxp  5465  tfrlemibfn  6437  tfr1onlembfn  6453  tfrcllembfn  6466  frecuzrdgtcl  10594  frecuzrdgdomlem  10599  dvbssntrcntop  15271