Theorem ffdm 5235

Description: A mapping is a partial function. (Contributed by set.mm contributors, 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 5227	. . . 4 |- (F:A-->B -> dom F = A)
2	1	feq2d 5216	. . 3 |- (F:A-->B -> (F:dom F-->B <-> F:A-->B))
3	2	ibir 233	. 2 |- (F:A-->B -> F:dom F-->B)
4		eqimss 3324	. . 3 |- (dom F = A -> dom F C_ A)
5	1, 4	syl 15	. 2 |- (F:A-->B -> dom F C_ A)
6	3, 5	jca 518	1 |- (F:A-->B -> (F:dom F-->B /\ dom F C_ A))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   C_ wss 3258  dom cdm 4773  -->wf 4778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-fn 4791  df-f 4792

This theorem is referenced by: (None)