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 478	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 590	. 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 717	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6583	. . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
11	9, 10	mpbiri 259	. 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
12	5, 11	impbii 210	1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)

Syntax hints:   ↔ wb 207   = wceq 1547  ∅c0 4268  dom cdm 5625  Rel wrel 5630  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 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-fun 6494  df-fn 6495

This theorem is referenced by:  mpt0  6634  f0  6715  f00  6716  f0bi  6717  f1o00  6809  fo00  6810  tpos0  8203  ixp0x  8871  0fz1  13496  hashf1  14417  fuchom  17929  grpinvfvi  18956  mulgfval  19043  mulgfvalALT  19044  mulgfvi  19047  0frgp  19752  invrfval  20367  psrvscafval  21930  tmdgsum  24085  deg1fvi  26075  hon0  31889  fconst7v  32719  fnchoice  45484  dvnprodlem3  46398  0funcg2  49581  0funcALT  49585  0fucterm  50040