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Theorem ffdm 5403
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 5393 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5380 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 233 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3230 . . 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 1623    C_ wss 3152   dom cdm 4689   -->wf 5251
This theorem is referenced by:  smoiso  6379  dfac21  27164  islindf2  27284  f1lindf  27292  s4f1o  28093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-fn 5258  df-f 5259
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