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Theorem ffdm 5419
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 5409 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5396 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 233 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3243 . . 3  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
51, 4syl 15 . 2  |-  ( F : A --> B  ->  dom  F  C_  A )
63, 5jca 518 1  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    C_ wss 3165   dom cdm 4705   -->wf 5267
This theorem is referenced by:  smoiso  6395  dfac21  27267  islindf2  27387  f1lindf  27395  s4f1o  28225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-fn 5274  df-f 5275
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