MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fn0 Structured version   Visualization version   GIF version

Theorem fn0 6630
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
StepHypRef Expression
1 fnrel 6602 . . 3 (𝐹 Fn ∅ → Rel 𝐹)
2 fndm 6603 . . 3 (𝐹 Fn ∅ → dom 𝐹 = ∅)
3 reldm0 5882 . . . 4 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
43biimpar 478 . . 3 ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
51, 2, 4syl2anc 584 . 2 (𝐹 Fn ∅ → 𝐹 = ∅)
6 fun0 6564 . . . 4 Fun ∅
7 dm0 5875 . . . 4 dom ∅ = ∅
8 df-fn 6497 . . . 4 (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
96, 7, 8mpbir2an 709 . . 3 ∅ Fn ∅
10 fneq1 6591 . . 3 (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
119, 10mpbiri 257 . 2 (𝐹 = ∅ → 𝐹 Fn ∅)
125, 11impbii 208 1 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  c0 4281  dom cdm 5632  Rel wrel 5637  Fun wfun 6488   Fn wfn 6489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-fun 6496  df-fn 6497
This theorem is referenced by:  mpt0  6641  f0  6721  f00  6722  f0bi  6723  f1o00  6817  fo00  6818  tpos0  8184  ixp0x  8861  0fz1  13458  hashf1  14353  fuchom  17846  fuchomOLD  17847  grpinvfvi  18790  mulgfval  18870  mulgfvalALT  18871  mulgfvi  18874  0frgp  19557  invrfval  20098  psrvscafval  21354  tmdgsum  23442  deg1fvi  25446  hon0  30633  fnchoice  43214  dvnprodlem3  44159
  Copyright terms: Public domain W3C validator