Theorem fn0 6678

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 6648	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6649	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5925	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 478	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 584	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6610	. . . 4 ⊢ Fun ∅
7		dm0 5918	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6543	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 709	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6637	. . 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 1541  ∅c0 4321  dom cdm 5675  Rel wrel 5680  Fun wfun 6534   Fn wfn 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-fun 6542  df-fn 6543

This theorem is referenced by:  mpt0  6689  f0  6769  f00  6770  f0bi  6771  f1o00  6865  fo00  6866  tpos0  8237  ixp0x  8916  0fz1  13517  hashf1  14414  fuchom  17909  fuchomOLD  17910  grpinvfvi  18863  mulgfval  18946  mulgfvalALT  18947  mulgfvi  18950  0frgp  19641  invrfval  20195  psrvscafval  21500  tmdgsum  23590  deg1fvi  25594  hon0  31033  fnchoice  43698  dvnprodlem3  44650