Theorem fn0 5336

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 5315	. . 3 |- ( F Fn (/) -> Rel F )
2		fndm 5316	. . 3 |- ( F Fn (/) -> dom F = (/) )
3		reldm0 4846	. . . 4 |- ( Rel F -> ( F = (/) <-> dom F = (/) ) )
4	3	biimpar 297	. . 3 |- ( ( Rel F /\ dom F = (/) ) -> F = (/) )
5	1, 2, 4	syl2anc 411	. 2 |- ( F Fn (/) -> F = (/) )
6		fun0 5275	. . . 4 |- Fun (/)
7		dm0 4842	. . . 4 |- dom (/) = (/)
8		df-fn 5220	. . . 4 |- ( (/) Fn (/) <-> ( Fun (/) /\ dom (/) = (/) ) )
9	6, 7, 8	mpbir2an 942	. . 3 |- (/) Fn (/)
10		fneq1 5305	. . 3 |- ( F = (/) -> ( F Fn (/) <-> (/) Fn (/) ) )
11	9, 10	mpbiri 168	. 2 |- ( F = (/) -> F Fn (/) )
12	5, 11	impbii 126	1 |- ( F Fn (/) <-> F = (/) )

Syntax hints:   <-> wb 105   = wceq 1353   (/)c0 3423   dom cdm 4627   Rel wrel 4632   Fun wfun 5211   Fn wfn 5212

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-fun 5219  df-fn 5220

This theorem is referenced by:  mpt0  5344  f0  5407  f00  5408  f0bi  5409  f1o00  5497  fo00  5498  tpos0  6275  ixp0x  6726  0fz1  10045