Theorem fn0 5924

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 5903	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 5904	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5264	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 501	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 691	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 5868	. . . 4 ⊢ Fun ∅
7		dm0 5260	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 5807	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 957	. . 3 ⊢ ∅ Fn ∅
10		fneq1 5893	. . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
11	9, 10	mpbiri 247	. 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
12	5, 11	impbii 198	1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)

Syntax hints:   ↔ wb 195   = wceq 1475  ∅c0 3874  dom cdm 5038  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807

This theorem is referenced by:  mpt0  5934  f0  5999  f00  6000  f0bi  6001  f1o00  6083  fo00  6084  tpos0  7269  ixp0x  7822  0fz1  12232  hashf1  13098  fuchom  16444  grpinvfvi  17286  mulgfval  17365  mulgfvi  17368  symgplusg  17632  0frgp  18015  invrfval  18496  psrvscafval  19211  tmdgsum  21709  deg1fvi  23649  hon0  28036  fnchoice  38211  dvnprodlem3  38838