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Theorem ffdm 5545
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 5535 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5521 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 234 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3343 . . 3  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
51, 4syl 16 . 2  |-  ( F : A --> B  ->  dom  F  C_  A )
63, 5jca 519 1  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    C_ wss 3263   dom cdm 4818   -->wf 5390
This theorem is referenced by:  smoiso  6560  s4f1o  11792  dfac21  26833  islindf2  26953  f1lindf  26961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-in 3270  df-ss 3277  df-fn 5397  df-f 5398
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