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Theorem fn0 5379
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 5358 . . 3  |-  ( F  Fn  (/)  ->  Rel  F )
2 fndm 5359 . . 3  |-  ( F  Fn  (/)  ->  dom  F  =  (/) )
3 reldm0 4912 . . . 4  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
43biimpar 471 . . 3  |-  ( ( Rel  F  /\  dom  F  =  (/) )  ->  F  =  (/) )
51, 2, 4syl2anc 642 . 2  |-  ( F  Fn  (/)  ->  F  =  (/) )
6 fun0 5323 . . . 4  |-  Fun  (/)
7 dm0 4908 . . . 4  |-  dom  (/)  =  (/)
8 df-fn 5274 . . . 4  |-  ( (/)  Fn  (/) 
<->  ( Fun  (/)  /\  dom  (/)  =  (/) ) )
96, 7, 8mpbir2an 886 . . 3  |-  (/)  Fn  (/)
10 fneq1 5349 . . 3  |-  ( F  =  (/)  ->  ( F  Fn  (/)  <->  (/)  Fn  (/) ) )
119, 10mpbiri 224 . 2  |-  ( F  =  (/)  ->  F  Fn  (/) )
125, 11impbii 180 1  |-  ( F  Fn  (/)  <->  F  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632   (/)c0 3468   dom cdm 4705   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266
This theorem is referenced by:  mpt0  5387  f0  5441  f00  5442  f1o00  5524  fo00  5525  fconstfv  5750  tpos0  6280  map0e  6821  ixp0x  6860  0fz1  10829  hashf1  11411  fuchom  13851  grpinvfvi  14539  mulgfval  14584  mulgfvi  14587  symgplusg  14792  0frgp  15104  invrfval  15471  psrvscafval  16151  tmdgsum  17794  deg1fvi  19487  hon0  22389  fnchoice  27803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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