MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ffdm Structured version   Unicode version

Theorem ffdm 5597
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 5587 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5573 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 234 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3392 . . 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 1652    C_ wss 3312   dom cdm 4870   -->wf 5442
This theorem is referenced by:  smoiso  6616  s4f1o  11857  dfac21  27132  islindf2  27252  f1lindf  27260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-in 3319  df-ss 3326  df-fn 5449  df-f 5450
  Copyright terms: Public domain W3C validator