Theorem fn0 6695

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 6666	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6667	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5918	. . . 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 6630	. . . 4 ⊢ Fun ∅
7		dm0 5910	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6564	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 724	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6655	. . 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 1570  ∅c0 4281  dom cdm 5656  Rel wrel 5661  Fun wfun 6555   Fn wfn 6556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-fun 6563  df-fn 6564

This theorem is used by:  mpt0  6706  f0  6788  f00  6789  f0bi  6790  f1o00  6886  fo00  6887  tpos0  8304  ixp0x  8985  0fz1  13707  hashf1  14632  fuchom  18171  grpinvfvi  19225  mulgfval  19311  mulgfvalALT  19312  mulgfvi  19315  0frgp  20026  invrfval  20652  psrvscafval  22289  tmdgsum  24447  deg1fvi  26436  hon0  32428  fconst7v  33289  fnchoice  46262  dvnprodlem3  47173  0funcg2  50414  0funcALT  50418  0fucterm  50896