Theorem ffdm 5081

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 5070	. . . 4 |- ( F : A --> B -> dom F = A )
2	1	feq2d 5055	. . 3 |- ( F : A --> B -> ( F : dom F --> B <-> F : A --> B ) )
3	2	ibir 175	. 2 |- ( F : A --> B -> F : dom F --> B )
4		eqimss 3051	. . 3 |- ( dom F = A -> dom F C_ A )
5	1, 4	syl 14	. 2 |- ( F : A --> B -> dom F C_ A )
6	3, 5	jca 300	1 |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   C_ wss 2973   dom cdm 4363   -->wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-fn 4925  df-f 4926

This theorem is referenced by:  smoiso  5940