Theorem fn0 6011

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 5989	. . 3 |- ( F Fn (/) -> Rel F )
2		fndm 5990	. . 3 |- ( F Fn (/) -> dom F = (/) )
3		reldm0 5343	. . . 4 |- ( Rel F -> ( F = (/) <-> dom F = (/) ) )
4	3	biimpar 502	. . 3 |- ( ( Rel F /\ dom F = (/) ) -> F = (/) )
5	1, 2, 4	syl2anc 693	. 2 |- ( F Fn (/) -> F = (/) )
6		fun0 5954	. . . 4 |- Fun (/)
7		dm0 5339	. . . 4 |- dom (/) = (/)
8		df-fn 5891	. . . 4 |- ( (/) Fn (/) <-> ( Fun (/) /\ dom (/) = (/) ) )
9	6, 7, 8	mpbir2an 955	. . 3 |- (/) Fn (/)
10		fneq1 5979	. . 3 |- ( F = (/) -> ( F Fn (/) <-> (/) Fn (/) ) )
11	9, 10	mpbiri 248	. 2 |- ( F = (/) -> F Fn (/) )
12	5, 11	impbii 199	1 |- ( F Fn (/) <-> F = (/) )

Syntax hints:   <-> wb 196   = wceq 1483   (/)c0 3915   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891

This theorem is referenced by:  mpt0  6021  f0  6086  f00  6087  f0bi  6088  f1o00  6171  fo00  6172  tpos0  7382  ixp0x  7936  0fz1  12361  hashf1  13241  fuchom  16621  grpinvfvi  17463  mulgfval  17542  mulgfvi  17545  symgplusg  17809  0frgp  18192  invrfval  18673  psrvscafval  19390  tmdgsum  21899  deg1fvi  23845  hon0  28652  fnchoice  39188  dvnprodlem3  40163