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Theorem dmmptss 5005
Description: The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
Hypothesis
Ref Expression
dmmpo.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
dmmptss  |-  dom  F  C_  A
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem dmmptss
StepHypRef Expression
1 dmmpo.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
21dmmpt 5004 . 2  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
3 ssrab2 3152 . 2  |-  { x  e.  A  |  B  e.  _V }  C_  A
42, 3eqsstri 3099 1  |-  dom  F  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   {crab 2397   _Vcvv 2660    C_ wss 3041    |-> cmpt 3959   dom cdm 4509
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  mptrcl  5471  fvmptssdm  5473  elfvmptrab1  5483  mptexg  5613  dmmpossx  6065  tposssxp  6114  lmrcl  12287  cnprcl2k  12302  isxms2  12548
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