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Theorem ffdm 5683
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 5674 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5658 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 242 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3519 . . 3  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
51, 4syl 16 . 2  |-  ( F : A --> B  ->  dom  F  C_  A )
63, 5jca 532 1  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    C_ wss 3439   dom cdm 4951   -->wf 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-in 3446  df-ss 3453  df-fn 5532  df-f 5533
This theorem is referenced by:  smoiso  6936  s4f1o  12649  islindf2  18371  f1lindf  18379  dfac21  29587
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