Theorem ffdm 6062

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

Step	Hyp	Ref	Expression
1		fdm 6051	. . . 4 |- ( F : A --> B -> dom F = A )
2	1	feq2d 6031	. . 3 |- ( F : A --> B -> ( F : dom F --> B <-> F : A --> B ) )
3	2	ibir 257	. 2 |- ( F : A --> B -> F : dom F --> B )
4		eqimss 3657	. . 3 |- ( dom F = A -> dom F C_ A )
5	1, 4	syl 17	. 2 |- ( F : A --> B -> dom F C_ A )
6	3, 5	jca 554	1 |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   C_ wss 3574   dom cdm 5114   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-fn 5891  df-f 5892

This theorem is referenced by:  ffdmd  6063  smoiso  7459  s4f1o  13663  islindf2  20153  f1lindf  20161  dfac21  37636  itgperiod  40197  fourierdlem92  40415  fouriersw  40448  etransclem2  40453