Theorem fn0 5282

Description: A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fn0	|- ( F Fn (/) <-> F = (/) )

Proof of Theorem fn0

Step	Hyp	Ref	Expression
1		fnrel 5261	. . 3 |- ( F Fn (/) -> Rel F )
2		fndm 5262	. . 3 |- ( F Fn (/) -> dom F = (/) )
3		reldm0 4797	. . . 4 |- ( Rel F -> ( F = (/) <-> dom F = (/) ) )
4	3	biimpar 295	. . 3 |- ( ( Rel F /\ dom F = (/) ) -> F = (/) )
5	1, 2, 4	syl2anc 409	. 2 |- ( F Fn (/) -> F = (/) )
6		fun0 5221	. . . 4 |- Fun (/)
7		dm0 4793	. . . 4 |- dom (/) = (/)
8		df-fn 5166	. . . 4 |- ( (/) Fn (/) <-> ( Fun (/) /\ dom (/) = (/) ) )
9	6, 7, 8	mpbir2an 927	. . 3 |- (/) Fn (/)
10		fneq1 5251	. . 3 |- ( F = (/) -> ( F Fn (/) <-> (/) Fn (/) ) )
11	9, 10	mpbiri 167	. 2 |- ( F = (/) -> F Fn (/) )
12	5, 11	impbii 125	1 |- ( F Fn (/) <-> F = (/) )

Syntax hints:   <-> wb 104   = wceq 1332   (/)c0 3390   dom cdm 4579   Rel wrel 4584   Fun wfun 5157   Fn wfn 5158

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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-fun 5165  df-fn 5166

This theorem is referenced by:  mpt0  5290  f0  5353  f00  5354  f0bi  5355  f1o00  5442  fo00  5443  tpos0  6211  ixp0x  6660  0fz1  9925