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Theorem ffdm 5368
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 5353 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5335 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 176 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3201 . . 3  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
51, 4syl 14 . 2  |-  ( F : A --> B  ->  dom  F  C_  A )
63, 5jca 304 1  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    C_ wss 3121   dom cdm 4611   -->wf 5194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-fn 5201  df-f 5202
This theorem is referenced by:  smoiso  6281  dvcj  13467  dvfre  13468
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