Theorem fn0 6666

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 6637	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6638	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5917	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 482	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 595	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6601	. . . 4 ⊢ Fun ∅
7		dm0 5909	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6539	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 723	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6626	. . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
11	9, 10	mpbiri 261	. 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
12	5, 11	impbii 212	1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∅c0 4285  dom cdm 5660  Rel wrel 5665  Fun wfun 6530   Fn wfn 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-fun 6538  df-fn 6539

This theorem is used by:  mpt0  6677  f0  6759  f00  6760  f0bi  6761  f1o00  6856  fo00  6857  tpos0  8250  ixp0x  8922  0fz1  13578  hashf1  14501  fuchom  18027  grpinvfvi  19055  mulgfval  19141  mulgfvalALT  19142  mulgfvi  19145  0frgp  19855  invrfval  20478  psrvscafval  22109  tmdgsum  24263  deg1fvi  26253  hon0  32156  fconst7v  32976  fnchoice  45777  dvnprodlem3  46690  0funcg2  49890  0funcALT  49894  0fucterm  50349