Theorem ffdm 4578

Description: A mapping is a partial function.

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 4567	. . . 4 |- (F:A-->B -> dom F = A)
2	1	feq2d 4557	. . 3 |- (F:A-->B -> (F:dom F-->B <-> F:A-->B))
3	2	ibir 653	. 2 |- (F:A-->B -> F:dom F-->B)
4		eqimss 2665	. . 3 |- (dom F = A -> dom F C_ A)
5	1, 4	syl 12	. 2 |- (F:A-->B -> dom F C_ A)
6	3, 5	jca 310	1 |- (F:A-->B -> (F:dom F-->B /\ dom F C_ A))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   C_ wss 2593  dom cdm 3986  -->wf 3994

This theorem is referenced by:  fpm 5397  smoiso 16453

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605  df-fn 4009  df-f 4010

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