Theorem fn0 6671

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 6642	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6643	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5909	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 483	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 596	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6606	. . . 4 ⊢ Fun ∅
7		dm0 5901	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6542	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 724	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6631	. . 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 4278  dom cdm 5650  Rel wrel 5655  Fun wfun 6533   Fn wfn 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-fun 6541  df-fn 6542

This theorem is used by:  mpt0  6682  f0  6764  f00  6765  f0bi  6766  f1o00  6861  fo00  6862  tpos0  8277  ixp0x  8958  0fz1  13681  hashf1  14606  fuchom  18145  grpinvfvi  19199  mulgfval  19285  mulgfvalALT  19286  mulgfvi  19289  0frgp  19999  invrfval  20625  psrvscafval  22262  tmdgsum  24420  deg1fvi  26409  hon0  32401  fconst7v  33262  fnchoice  46140  dvnprodlem3  47052  0funcg2  50293  0funcALT  50297  0fucterm  50775