Theorem ffdm 5431

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 5416	. . . 4 |- ( F : A --> B -> dom F = A )
2	1	feq2d 5398	. . 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 3238	. . 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 1364   C_ wss 3157   dom cdm 4664   -->wf 5255

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-fn 5262  df-f 5263

This theorem is referenced by:  smoiso  6369  dvcj  15029  dvfre  15030