Theorem fn0 6481

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 6456	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6457	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5800	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 480	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 586	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6421	. . . 4 ⊢ Fun ∅
7		dm0 5792	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6360	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 709	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6446	. . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
11	9, 10	mpbiri 260	. 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
12	5, 11	impbii 211	1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)

Syntax hints:   ↔ wb 208   = wceq 1537  ∅c0 4293  dom cdm 5557  Rel wrel 5562  Fun wfun 6351   Fn wfn 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-fun 6359  df-fn 6360

This theorem is referenced by:  mpt0  6492  f0  6562  f00  6563  f0bi  6564  f1o00  6651  fo00  6652  tpos0  7924  ixp0x  8492  0fz1  12930  hashf1  13818  fuchom  17233  grpinvfvi  18148  mulgfval  18228  mulgfvalALT  18229  mulgfvi  18232  0frgp  18907  invrfval  19425  psrvscafval  20172  tmdgsum  22705  deg1fvi  24681  hon0  29572  fnchoice  41293  dvnprodlem3  42240