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Theorem ffdm 5339
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 5324 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5306 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 176 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3182 . . 3  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
51, 4syl 14 . 2  |-  ( F : A --> B  ->  dom  F  C_  A )
63, 5jca 304 1  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    C_ wss 3102   dom cdm 4585   -->wf 5165
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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115  df-fn 5172  df-f 5173
This theorem is referenced by:  smoiso  6246  dvcj  13060  dvfre  13061
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