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

Theorem dmss 4602
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 3008 . . . 4  |-  ( A 
C_  B  ->  ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
) )
21eximdv 1805 . . 3  |-  ( A 
C_  B  ->  ( E. y <. x ,  y
>.  e.  A  ->  E. y <. x ,  y >.  e.  B ) )
3 vex 2618 . . . 4  |-  x  e. 
_V
43eldm2 4601 . . 3  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
53eldm2 4601 . . 3  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
62, 4, 53imtr4g 203 . 2  |-  ( A 
C_  B  ->  (
x  e.  dom  A  ->  x  e.  dom  B
) )
76ssrdv 3020 1  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1424    e. wcel 1436    C_ wss 2988   <.cop 3434   dom cdm 4410
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-dm 4420
This theorem is referenced by:  dmeq  4603  dmv  4619  rnss  4632  dmiin  4648  dmxpss2  4826  ssxpbm  4829  ssxp1  4830  cocnvres  4918  relrelss  4920  funssxp  5138  fvun1  5327  fndmdif  5361  fneqeql2  5365  tposss  5959  smores  6005  smores2  6007  tfrlemibfn  6041  tfrlemiubacc  6043  tfr1onlembfn  6057  tfr1onlemubacc  6059  tfr1onlemres  6062  tfrcllembfn  6070  tfrcllemubacc  6072  tfrcllemres  6075  frecuzrdgtcl  9740  frecuzrdgdomlem  9745
  Copyright terms: Public domain W3C validator