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

Theorem dmss 4738
Description: Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
Assertion
Ref Expression
dmss  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )

Proof of Theorem dmss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3091 . . . 4  |-  ( A 
C_  B  ->  ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
) )
21eximdv 1852 . . 3  |-  ( A 
C_  B  ->  ( E. y <. x ,  y
>.  e.  A  ->  E. y <. x ,  y >.  e.  B ) )
3 vex 2689 . . . 4  |-  x  e. 
_V
43eldm2 4737 . . 3  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
53eldm2 4737 . . 3  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
62, 4, 53imtr4g 204 . 2  |-  ( A 
C_  B  ->  (
x  e.  dom  A  ->  x  e.  dom  B
) )
76ssrdv 3103 1  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1468    e. wcel 1480    C_ wss 3071   <.cop 3530   dom cdm 4539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-dm 4549
This theorem is referenced by:  dmeq  4739  dmv  4755  rnss  4769  dmiin  4785  dmxpss2  4971  ssxpbm  4974  ssxp1  4975  cocnvres  5063  relrelss  5065  funssxp  5292  fvun1  5487  fndmdif  5525  fneqeql2  5529  tposss  6143  smores  6189  smores2  6191  tfrlemibfn  6225  tfrlemiubacc  6227  tfr1onlembfn  6241  tfr1onlemubacc  6243  tfr1onlemres  6246  tfrcllembfn  6254  tfrcllemubacc  6256  tfrcllemres  6259  frecuzrdgtcl  10192  frecuzrdgdomlem  10197  ennnfonelemex  11934  strleund  12057  strleun  12058  dvbssntrcntop  12832
  Copyright terms: Public domain W3C validator