Theorem fn0 6548

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 6519	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6520	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5826	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 477	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 583	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6483	. . . 4 ⊢ Fun ∅
7		dm0 5818	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6421	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 707	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6508	. . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
11	9, 10	mpbiri 257	. 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
12	5, 11	impbii 208	1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)

Syntax hints:   ↔ wb 205   = wceq 1539  ∅c0 4253  dom cdm 5580  Rel wrel 5585  Fun wfun 6412   Fn wfn 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-fun 6420  df-fn 6421

This theorem is referenced by:  mpt0  6559  f0  6639  f00  6640  f0bi  6641  f1o00  6734  fo00  6735  tpos0  8043  ixp0x  8672  0fz1  13205  hashf1  14099  fuchom  17594  fuchomOLD  17595  grpinvfvi  18537  mulgfval  18617  mulgfvalALT  18618  mulgfvi  18621  0frgp  19300  invrfval  19830  psrvscafval  21069  tmdgsum  23154  deg1fvi  25155  hon0  30056  fnchoice  42461  dvnprodlem3  43379