Theorem fn0 5443

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 5419	. . 3 |- ( F Fn (/) -> Rel F )
2		fndm 5420	. . 3 |- ( F Fn (/) -> dom F = (/) )
3		reldm0 4941	. . . 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 5379	. . . 4 |- Fun (/)
7		dm0 4937	. . . 4 |- dom (/) = (/)
8		df-fn 5321	. . . 4 |- ( (/) Fn (/) <-> ( Fun (/) /\ dom (/) = (/) ) )
9	6, 7, 8	mpbir2an 948	. . 3 |- (/) Fn (/)
10		fneq1 5409	. . 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 1395   (/)c0 3491   dom cdm 4719   Rel wrel 4724   Fun wfun 5312   Fn wfn 5313

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-fun 5320  df-fn 5321

This theorem is referenced by:  mpt0  5451  f0  5518  f00  5519  f0bi  5520  f1o00  5610  fo00  5611  tpos0  6426  ixp0x  6881  0fz1  10253