ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmxpss Unicode version

Theorem dmxpss 4777
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.
StepHypRef Expression
1 vex 2605 . . . 4  |-  x  e. 
_V
21eldm2 4555 . . 3  |-  ( x  e.  dom  ( A  X.  B )  <->  E. y <. x ,  y >.  e.  ( A  X.  B
) )
3 opelxp1 4397 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  e.  A )
43exlimiv 1530 . . 3  |-  ( E. y <. x ,  y
>.  e.  ( A  X.  B )  ->  x  e.  A )
52, 4sylbi 119 . 2  |-  ( x  e.  dom  ( A  X.  B )  ->  x  e.  A )
65ssriv 3004 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff set class
Syntax hints:   E.wex 1422    e. wcel 1434    C_ wss 2974   <.cop 3403    X. cxp 4363   dom cdm 4365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-dm 4375
This theorem is referenced by:  rnxpss  4778  ssxpbm  4780  ssxp1  4781  funssxp  5085  tfrlemibfn  5971  tfr1onlembfn  5987  tfrcllembfn  6000  frecuzrdgtcl  9483  frecuzrdgdomlem  9488
  Copyright terms: Public domain W3C validator