Theorem fn0 6612

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 6583	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6584	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5868	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 477	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 584	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6546	. . . 4 ⊢ Fun ∅
7		dm0 5860	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6484	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 711	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6572	. . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
11	9, 10	mpbiri 258	. 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
12	5, 11	impbii 209	1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1541  ∅c0 4283  dom cdm 5616  Rel wrel 5621  Fun wfun 6475   Fn wfn 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-fun 6483  df-fn 6484

This theorem is referenced by:  mpt0  6623  f0  6704  f00  6705  f0bi  6706  f1o00  6798  fo00  6799  tpos0  8186  ixp0x  8850  0fz1  13441  hashf1  14361  fuchom  17868  grpinvfvi  18892  mulgfval  18979  mulgfvalALT  18980  mulgfvi  18983  0frgp  19689  invrfval  20305  psrvscafval  21883  tmdgsum  24008  deg1fvi  26015  hon0  31768  fconst7v  32598  fnchoice  45065  dvnprodlem3  45985  0funcg2  49115  0funcALT  49119  0fucterm  49574