Theorem fn0 5451

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 5427	. . 3 |- ( F Fn (/) -> Rel F )
2		fndm 5428	. . 3 |- ( F Fn (/) -> dom F = (/) )
3		reldm0 4948	. . . 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 5387	. . . 4 |- Fun (/)
7		dm0 4944	. . . 4 |- dom (/) = (/)
8		df-fn 5328	. . . 4 |- ( (/) Fn (/) <-> ( Fun (/) /\ dom (/) = (/) ) )
9	6, 7, 8	mpbir2an 950	. . 3 |- (/) Fn (/)
10		fneq1 5417	. . 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 1397   (/)c0 3493   dom cdm 4724   Rel wrel 4729   Fun wfun 5319   Fn wfn 5320

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-br 4088  df-opab 4150  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-fun 5327  df-fn 5328

This theorem is referenced by:  mpt0  5459  f0  5527  f00  5528  f0bi  5529  f1o00  5620  fo00  5621  tpos0  6442  ixp0x  6897  0fz1  10282