Theorem fn0 6669

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 6640	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6641	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5907	. . . 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 6601	. . . 4 ⊢ Fun ∅
7		dm0 5900	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6534	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 711	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6629	. . 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 4308  dom cdm 5654  Rel wrel 5659  Fun wfun 6525   Fn wfn 6526

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-fun 6533  df-fn 6534

This theorem is referenced by:  mpt0  6680  f0  6759  f00  6760  f0bi  6761  f1o00  6853  fo00  6854  tpos0  8255  ixp0x  8940  0fz1  13561  hashf1  14475  fuchom  17977  grpinvfvi  18965  mulgfval  19052  mulgfvalALT  19053  mulgfvi  19056  0frgp  19760  invrfval  20349  psrvscafval  21908  tmdgsum  24033  deg1fvi  26042  hon0  31774  fnchoice  45053  dvnprodlem3  45977  0funcg2  49049  0funcALT  49053