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Theorem dmss 4838
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 3161 . . . 4  |-  ( A 
C_  B  ->  ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
) )
21eximdv 1890 . . 3  |-  ( A 
C_  B  ->  ( E. y <. x ,  y
>.  e.  A  ->  E. y <. x ,  y >.  e.  B ) )
3 vex 2752 . . . 4  |-  x  e. 
_V
43eldm2 4837 . . 3  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
53eldm2 4837 . . 3  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
62, 4, 53imtr4g 205 . 2  |-  ( A 
C_  B  ->  (
x  e.  dom  A  ->  x  e.  dom  B
) )
76ssrdv 3173 1  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1502    e. wcel 2158    C_ wss 3141   <.cop 3607   dom cdm 4638
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-dm 4648
This theorem is referenced by:  dmeq  4839  dmv  4855  rnss  4869  dmiin  4885  dmxpss2  5073  ssxpbm  5076  ssxp1  5077  cocnvres  5165  relrelss  5167  funssxp  5397  fvun1  5595  fndmdif  5634  fneqeql2  5638  tposss  6261  smores  6307  smores2  6309  tfrlemibfn  6343  tfrlemiubacc  6345  tfr1onlembfn  6359  tfr1onlemubacc  6361  tfr1onlemres  6364  tfrcllembfn  6372  tfrcllemubacc  6374  tfrcllemres  6377  frecuzrdgtcl  10426  frecuzrdgdomlem  10431  ennnfonelemex  12429  strleund  12577  strleun  12578  imasaddfnlemg  12753  dvbssntrcntop  14449
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