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Theorem ffdm 5386
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 5371 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5353 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 177 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3209 . . 3  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
51, 4syl 14 . 2  |-  ( F : A --> B  ->  dom  F  C_  A )
63, 5jca 306 1  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    C_ wss 3129   dom cdm 4626   -->wf 5212
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142  df-fn 5219  df-f 5220
This theorem is referenced by:  smoiso  6302  dvcj  14066  dvfre  14067
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