Theorem fn0 6617

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 6588	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6589	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5872	. . . 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 6551	. . . 4 ⊢ Fun ∅
7		dm0 5864	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6489	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 711	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6577	. . 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 4282  dom cdm 5619  Rel wrel 5624  Fun wfun 6480   Fn wfn 6481

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-fun 6488  df-fn 6489

This theorem is referenced by:  mpt0  6628  f0  6709  f00  6710  f0bi  6711  f1o00  6803  fo00  6804  tpos0  8192  ixp0x  8856  0fz1  13446  hashf1  14366  fuchom  17873  grpinvfvi  18897  mulgfval  18984  mulgfvalALT  18985  mulgfvi  18988  0frgp  19693  invrfval  20309  psrvscafval  21887  tmdgsum  24011  deg1fvi  26018  hon0  31775  fconst7v  32605  fnchoice  45150  dvnprodlem3  46070  0funcg2  49209  0funcALT  49213  0fucterm  49668