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Theorem dmmptss 5137
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 5136 . 2  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
3 ssrab2 3252 . 2  |-  { x  e.  A  |  B  e.  _V }  C_  A
42, 3eqsstri 3199 1  |-  dom  F  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1363    e. wcel 2158   {crab 2469   _Vcvv 2749    C_ wss 3141    |-> cmpt 4076   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-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-mpt 4078  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651
This theorem is referenced by:  mptrcl  5611  fvmptssdm  5613  elfvmptrab1  5623  mptexg  5754  mptexw  6128  dmmpossx  6214  tposssxp  6264  lmrcl  14044  cnprcl2k  14059  isxms2  14305
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