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Theorem fssdm 5402
Description: Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, semi-deduction form. (Contributed by AV, 21-Aug-2022.)
Hypotheses
Ref Expression
fssdm.d  |-  D  C_  dom  F
fssdm.f  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
fssdm  |-  ( ph  ->  D  C_  A )

Proof of Theorem fssdm
StepHypRef Expression
1 fssdm.d . 2  |-  D  C_  dom  F
2 fssdm.f . . 3  |-  ( ph  ->  F : A --> B )
32fdmd 5394 . 2  |-  ( ph  ->  dom  F  =  A )
41, 3sseqtrid 3220 1  |-  ( ph  ->  D  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3144   dom cdm 4647   -->wf 5234
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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-fn 5241  df-f 5242
This theorem is referenced by:  fisumss  11441  fprodssdc  11639  ghmpreima  13230  cnclima  14208  txcnmpt  14258  xmeter  14421
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