Theorem fn0 5043

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 5022	. . 3 |- ( F Fn (/) -> Rel F )
2		fndm 5023	. . 3 |- ( F Fn (/) -> dom F = (/) )
3		reldm0 4575	. . . 4 |- ( Rel F -> ( F = (/) <-> dom F = (/) ) )
4	3	biimpar 291	. . 3 |- ( ( Rel F /\ dom F = (/) ) -> F = (/) )
5	1, 2, 4	syl2anc 403	. 2 |- ( F Fn (/) -> F = (/) )
6		fun0 4982	. . . 4 |- Fun (/)
7		dm0 4571	. . . 4 |- dom (/) = (/)
8		df-fn 4929	. . . 4 |- ( (/) Fn (/) <-> ( Fun (/) /\ dom (/) = (/) ) )
9	6, 7, 8	mpbir2an 884	. . 3 |- (/) Fn (/)
10		fneq1 5012	. . 3 |- ( F = (/) -> ( F Fn (/) <-> (/) Fn (/) ) )
11	9, 10	mpbiri 166	. 2 |- ( F = (/) -> F Fn (/) )
12	5, 11	impbii 124	1 |- ( F Fn (/) <-> F = (/) )

Syntax hints:   <-> wb 103   = wceq 1285   (/)c0 3252   dom cdm 4365   Rel wrel 4370   Fun wfun 4920   Fn wfn 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-fun 4928  df-fn 4929

This theorem is referenced by:  mpt0  5051  f0  5105  f00  5106  f1o00  5186  fo00  5187  tpos0  5917  0fz1  9129