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Theorem dmmptss 4847
Description: The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
Hypothesis
Ref Expression
dmmpt2.1  |-  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 dmmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
21dmmpt 4846 . 2  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
3 ssrab2 3080 . 2  |-  { x  e.  A  |  B  e.  _V }  C_  A
42, 3eqsstri 3030 1  |-  dom  F  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   {crab 2353   _Vcvv 2602    C_ wss 2974    |-> cmpt 3847   dom cdm 4371
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-mpt 3849  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384
This theorem is referenced by:  fvmptssdm  5287  mptexg  5418  dmmpt2ssx  5856  tposssxp  5898
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