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

Theorem ffdm 5751
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 5741 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
21feq2d 5724 . . 3  |-  ( F : A --> B  -> 
( F : dom  F --> B  <->  F : A --> B ) )
32ibir 242 . 2  |-  ( F : A --> B  ->  F : dom  F --> B )
4 eqimss 3551 . . 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 1395    C_ wss 3471   dom cdm 5008   -->wf 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-in 3478  df-ss 3485  df-fn 5597  df-f 5598
This theorem is referenced by:  smoiso  7051  s4f1o  12878  islindf2  18976  f1lindf  18984  dfac21  31216  itgperiod  31983  fourierdlem92  32184  fouriersw  32217  etransclem2  32222
  Copyright terms: Public domain W3C validator