Theorem fn0 6648

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 6619	. . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹)
2		fndm 6620	. . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅)
3		reldm0 5902	. . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
4	3	biimpar 481	. . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
5	1, 2, 4	syl2anc 593	. 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
6		fun0 6582	. . . 4 ⊢ Fun ∅
7		dm0 5894	. . . 4 ⊢ dom ∅ = ∅
8		df-fn 6520	. . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
9	6, 7, 8	mpbir2an 721	. . 3 ⊢ ∅ Fn ∅
10		fneq1 6608	. . 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 1559  ∅c0 4285  dom cdm 5645  Rel wrel 5650  Fun wfun 6511   Fn wfn 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-fun 6519  df-fn 6520

This theorem is referenced by:  mpt0  6659  f0  6741  f00  6742  f0bi  6743  f1o00  6838  fo00  6839  tpos0  8231  ixp0x  8904  0fz1  13546  hashf1  14467  fuchom  17980  grpinvfvi  19007  mulgfval  19094  mulgfvalALT  19095  mulgfvi  19098  0frgp  19802  invrfval  20417  psrvscafval  21980  tmdgsum  24135  deg1fvi  26125  hon0  31942  fconst7v  32772  fnchoice  45573  dvnprodlem3  46486  0funcg2  49669  0funcALT  49673  0fucterm  50128