Theorem ffdm 5388

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 5373	. . . 4 |- ( F : A --> B -> dom F = A )
2	1	feq2d 5355	. . 3 |- ( F : A --> B -> ( F : dom F --> B <-> F : A --> B ) )
3	2	ibir 177	. 2 |- ( F : A --> B -> F : dom F --> B )
4		eqimss 3211	. . 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 306	1 |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   C_ wss 3131   dom cdm 4628   -->wf 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-fn 5221  df-f 5222

This theorem is referenced by:  smoiso  6305  dvcj  14258  dvfre  14259