Theorem fn0 6682

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 6652	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6653	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5928	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 479	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 585	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6614	. . . 4 ⊢ Fun ∅
7		dm0 5921	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6547	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 710	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6641	. . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
11	9, 10	mpbiri 258	. 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
12	5, 11	impbii 208	1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)

Syntax hints:   ↔ wb 205   = wceq 1542  ∅c0 4323  dom cdm 5677  Rel wrel 5682  Fun wfun 6538   Fn wfn 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-fun 6546  df-fn 6547

This theorem is referenced by:  mpt0  6693  f0  6773  f00  6774  f0bi  6775  f1o00  6869  fo00  6870  tpos0  8241  ixp0x  8920  0fz1  13521  hashf1  14418  fuchom  17913  fuchomOLD  17914  grpinvfvi  18867  mulgfval  18952  mulgfvalALT  18953  mulgfvi  18956  0frgp  19647  invrfval  20203  psrvscafval  21509  tmdgsum  23599  deg1fvi  25603  hon0  31046  fnchoice  43713  dvnprodlem3  44664