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Theorem fn0 5506
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 5485 . . 3  |-  ( F  Fn  (/)  ->  Rel  F )
2 fndm 5486 . . 3  |-  ( F  Fn  (/)  ->  dom  F  =  (/) )
3 reldm0 5029 . . . 4  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
43biimpar 472 . . 3  |-  ( ( Rel  F  /\  dom  F  =  (/) )  ->  F  =  (/) )
51, 2, 4syl2anc 643 . 2  |-  ( F  Fn  (/)  ->  F  =  (/) )
6 fun0 5450 . . . 4  |-  Fun  (/)
7 dm0 5025 . . . 4  |-  dom  (/)  =  (/)
8 df-fn 5399 . . . 4  |-  ( (/)  Fn  (/) 
<->  ( Fun  (/)  /\  dom  (/)  =  (/) ) )
96, 7, 8mpbir2an 887 . . 3  |-  (/)  Fn  (/)
10 fneq1 5476 . . 3  |-  ( F  =  (/)  ->  ( F  Fn  (/)  <->  (/)  Fn  (/) ) )
119, 10mpbiri 225 . 2  |-  ( F  =  (/)  ->  F  Fn  (/) )
125, 11impbii 181 1  |-  ( F  Fn  (/)  <->  F  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   (/)c0 3573   dom cdm 4820   Rel wrel 4825   Fun wfun 5390    Fn wfn 5391
This theorem is referenced by:  mpt0  5514  f0  5569  f00  5570  f1o00  5652  fo00  5653  fconstfv  5895  tpos0  6447  map0e  6989  ixp0x  7028  0fz1  11008  hashf1  11635  fuchom  14087  grpinvfvi  14775  mulgfval  14820  mulgfvi  14823  symgplusg  15028  0frgp  15340  invrfval  15707  psrvscafval  16383  tmdgsum  18048  deg1fvi  19877  hon0  23146  fnchoice  27370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-fun 5398  df-fn 5399
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