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

Theorem fn0 5706
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 5685 . . 3  |-  ( F  Fn  (/)  ->  Rel  F )
2 fndm 5686 . . 3  |-  ( F  Fn  (/)  ->  dom  F  =  (/) )
3 reldm0 5226 . . . 4  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
43biimpar 485 . . 3  |-  ( ( Rel  F  /\  dom  F  =  (/) )  ->  F  =  (/) )
51, 2, 4syl2anc 661 . 2  |-  ( F  Fn  (/)  ->  F  =  (/) )
6 fun0 5651 . . . 4  |-  Fun  (/)
7 dm0 5222 . . . 4  |-  dom  (/)  =  (/)
8 df-fn 5597 . . . 4  |-  ( (/)  Fn  (/) 
<->  ( Fun  (/)  /\  dom  (/)  =  (/) ) )
96, 7, 8mpbir2an 918 . . 3  |-  (/)  Fn  (/)
10 fneq1 5675 . . 3  |-  ( F  =  (/)  ->  ( F  Fn  (/)  <->  (/)  Fn  (/) ) )
119, 10mpbiri 233 . 2  |-  ( F  =  (/)  ->  F  Fn  (/) )
125, 11impbii 188 1  |-  ( F  Fn  (/)  <->  F  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379   (/)c0 3790   dom cdm 5005   Rel wrel 5010   Fun wfun 5588    Fn wfn 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-fun 5596  df-fn 5597
This theorem is referenced by:  mpt0  5714  f0  5772  f00  5773  f0bi  5774  f1o00  5854  fo00  5855  fconstfv  6134  tpos0  6997  map0e  7468  ixp0x  7509  0fz1  11717  hashf1  12487  fuchom  15205  grpinvfvi  15963  mulgfval  16015  mulgfvi  16018  symgplusg  16286  0frgp  16670  invrfval  17194  psrvscafval  17913  tmdgsum  20462  deg1fvi  22353  hon0  26535  fnchoice  31306
  Copyright terms: Public domain W3C validator