Theorem ffdm 5358

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 5343	. . . 4 |- ( F : A --> B -> dom F = A )
2	1	feq2d 5325	. . 3 |- ( F : A --> B -> ( F : dom F --> B <-> F : A --> B ) )
3	2	ibir 176	. 2 |- ( F : A --> B -> F : dom F --> B )
4		eqimss 3196	. . 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 304	1 |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   C_ wss 3116   dom cdm 4604   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-fn 5191  df-f 5192

This theorem is referenced by:  smoiso  6270  dvcj  13313  dvfre  13314