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Theorem dmss 4648
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 3020 . . . 4  |-  ( A 
C_  B  ->  ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
) )
21eximdv 1809 . . 3  |-  ( A 
C_  B  ->  ( E. y <. x ,  y
>.  e.  A  ->  E. y <. x ,  y >.  e.  B ) )
3 vex 2623 . . . 4  |-  x  e. 
_V
43eldm2 4647 . . 3  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
53eldm2 4647 . . 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 3032 1  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1427    e. wcel 1439    C_ wss 3000   <.cop 3453   dom cdm 4451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-dm 4461
This theorem is referenced by:  dmeq  4649  dmv  4665  rnss  4678  dmiin  4694  dmxpss2  4876  ssxpbm  4879  ssxp1  4880  cocnvres  4968  relrelss  4970  funssxp  5193  fvun1  5383  fndmdif  5418  fneqeql2  5422  tposss  6025  smores  6071  smores2  6073  tfrlemibfn  6107  tfrlemiubacc  6109  tfr1onlembfn  6123  tfr1onlemubacc  6125  tfr1onlemres  6128  tfrcllembfn  6136  tfrcllemubacc  6138  tfrcllemres  6141  frecuzrdgtcl  9873  frecuzrdgdomlem  9878  strleund  11636  strleun  11637
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