Theorem fn0 6629

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 6600	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6601	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5883	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 477	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 585	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6563	. . . 4 ⊢ Fun ∅
7		dm0 5875	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6501	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 712	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6589	. . 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 1542  ∅c0 4273  dom cdm 5631  Rel wrel 5636  Fun wfun 6492   Fn wfn 6493

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-fun 6500  df-fn 6501

This theorem is referenced by:  mpt0  6640  f0  6721  f00  6722  f0bi  6723  f1o00  6815  fo00  6816  tpos0  8206  ixp0x  8874  0fz1  13498  hashf1  14419  fuchom  17931  grpinvfvi  18958  mulgfval  19045  mulgfvalALT  19046  mulgfvi  19049  0frgp  19754  invrfval  20369  psrvscafval  21927  tmdgsum  24060  deg1fvi  26050  hon0  31864  fconst7v  32693  fnchoice  45460  dvnprodlem3  46376  0funcg2  49559  0funcALT  49563  0fucterm  50018