Theorem ffdm 5339

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 5324	. . . 4 |- ( F : A --> B -> dom F = A )
2	1	feq2d 5306	. . 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 3182	. . 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 1335   C_ wss 3102   dom cdm 4585   -->wf 5165

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115  df-fn 5172  df-f 5173

This theorem is referenced by:  smoiso  6246  dvcj  13060  dvfre  13061