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Theorem dmss 4562
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 2967 . . . 4  |-  ( A 
C_  B  ->  ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
) )
21eximdv 1776 . . 3  |-  ( A 
C_  B  ->  ( E. y <. x ,  y
>.  e.  A  ->  E. y <. x ,  y >.  e.  B ) )
3 vex 2577 . . . 4  |-  x  e. 
_V
43eldm2 4561 . . 3  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
53eldm2 4561 . . 3  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
62, 4, 53imtr4g 198 . 2  |-  ( A 
C_  B  ->  (
x  e.  dom  A  ->  x  e.  dom  B
) )
76ssrdv 2979 1  |-  ( A 
C_  B  ->  dom  A 
C_  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1397    e. wcel 1409    C_ wss 2945   <.cop 3406   dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-dm 4383
This theorem is referenced by:  dmeq  4563  dmv  4579  rnss  4592  dmiin  4608  ssxpbm  4784  ssxp1  4785  relrelss  4872  funssxp  5088  fvun1  5267  fndmdif  5300  fneqeql2  5304  tposss  5892  smores  5938  smores2  5940  tfrlemibfn  5973  tfrlemiubacc  5975  frecuzrdgfn  9362
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