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Theorem ffdm 5086
Description: A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
Assertion
Ref Expression
ffdm  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )

Proof of Theorem ffdm
StepHypRef Expression
1 fdm 5075 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5060 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 175 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3052 . . 3  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
51, 4syl 14 . 2  |-  ( F : A --> B  ->  dom  F  C_  A )
63, 5jca 300 1  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    C_ wss 2974   dom cdm 4365   -->wf 4922
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987  df-fn 4929  df-f 4930
This theorem is referenced by:  smoiso  5945
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