Theorem fn0 6623

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 6594	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6595	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5877	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 477	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 584	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6557	. . . 4 ⊢ Fun ∅
7		dm0 5869	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6495	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 711	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6583	. . 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 1541  ∅c0 4285  dom cdm 5624  Rel wrel 5629  Fun wfun 6486   Fn wfn 6487

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-fun 6494  df-fn 6495

This theorem is referenced by:  mpt0  6634  f0  6715  f00  6716  f0bi  6717  f1o00  6809  fo00  6810  tpos0  8198  ixp0x  8864  0fz1  13460  hashf1  14380  fuchom  17888  grpinvfvi  18912  mulgfval  18999  mulgfvalALT  19000  mulgfvi  19003  0frgp  19708  invrfval  20325  psrvscafval  21904  tmdgsum  24039  deg1fvi  26046  hon0  31868  fconst7v  32698  fnchoice  45270  dvnprodlem3  46188  0funcg2  49325  0funcALT  49329  0fucterm  49784