Theorem fn0 6049

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 6027	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6028	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5375	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 501	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 694	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 5992	. . . 4 ⊢ Fun ∅
7		dm0 5371	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 5929	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 975	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6017	. . 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 1523  ∅c0 3948  dom cdm 5143  Rel wrel 5148  Fun wfun 5920   Fn wfn 5921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-fun 5928  df-fn 5929

This theorem is referenced by:  mpt0  6059  f0  6124  f00  6125  f0bi  6126  f1o00  6209  fo00  6210  tpos0  7427  ixp0x  7978  0fz1  12399  hashf1  13279  fuchom  16668  grpinvfvi  17510  mulgfval  17589  mulgfvi  17592  symgplusg  17855  0frgp  18238  invrfval  18719  psrvscafval  19438  tmdgsum  21946  deg1fvi  23890  hon0  28780  fnchoice  39502  dvnprodlem3  40481