Theorem fn0 5250

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 5229	. . 3 |- ( F Fn (/) -> Rel F )
2		fndm 5230	. . 3 |- ( F Fn (/) -> dom F = (/) )
3		reldm0 4765	. . . 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 5189	. . . 4 |- Fun (/)
7		dm0 4761	. . . 4 |- dom (/) = (/)
8		df-fn 5134	. . . 4 |- ( (/) Fn (/) <-> ( Fun (/) /\ dom (/) = (/) ) )
9	6, 7, 8	mpbir2an 927	. . 3 |- (/) Fn (/)
10		fneq1 5219	. . 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 3368   dom cdm 4547   Rel wrel 4552   Fun wfun 5125   Fn wfn 5126

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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-fun 5133  df-fn 5134

This theorem is referenced by:  mpt0  5258  f0  5321  f00  5322  f0bi  5323  f1o00  5410  fo00  5411  tpos0  6179  ixp0x  6628  0fz1  9856