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 5874	. . . 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 5867	. . . 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 1540  ∅c0 4286  dom cdm 5623  Rel wrel 5628  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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-fun 6488  df-fn 6489

This theorem is referenced by:  mpt0  6628  f0  6709  f00  6710  f0bi  6711  f1o00  6803  fo00  6804  tpos0  8196  ixp0x  8860  0fz1  13465  hashf1  14382  fuchom  17889  grpinvfvi  18879  mulgfval  18966  mulgfvalALT  18967  mulgfvi  18970  0frgp  19676  invrfval  20292  psrvscafval  21873  tmdgsum  23998  deg1fvi  26006  hon0  31755  fnchoice  45007  dvnprodlem3  45930  0funcg2  49070  0funcALT  49074  0fucterm  49529