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Theorem fn0 5556
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 5535 . . 3  |-  ( F  Fn  (/)  ->  Rel  F )
2 fndm 5536 . . 3  |-  ( F  Fn  (/)  ->  dom  F  =  (/) )
3 reldm0 5079 . . . 4  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
43biimpar 472 . . 3  |-  ( ( Rel  F  /\  dom  F  =  (/) )  ->  F  =  (/) )
51, 2, 4syl2anc 643 . 2  |-  ( F  Fn  (/)  ->  F  =  (/) )
6 fun0 5500 . . . 4  |-  Fun  (/)
7 dm0 5075 . . . 4  |-  dom  (/)  =  (/)
8 df-fn 5449 . . . 4  |-  ( (/)  Fn  (/) 
<->  ( Fun  (/)  /\  dom  (/)  =  (/) ) )
96, 7, 8mpbir2an 887 . . 3  |-  (/)  Fn  (/)
10 fneq1 5526 . . 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 1652   (/)c0 3620   dom cdm 4870   Rel wrel 4875   Fun wfun 5440    Fn wfn 5441
This theorem is referenced by:  mpt0  5564  f0  5619  f00  5620  f1o00  5702  fo00  5703  fconstfv  5946  tpos0  6501  map0e  7043  ixp0x  7082  0fz1  11066  hashf1  11698  fuchom  14150  grpinvfvi  14838  mulgfval  14883  mulgfvi  14886  symgplusg  15091  0frgp  15403  invrfval  15770  psrvscafval  16446  tmdgsum  18117  deg1fvi  20000  hon0  23288  fnchoice  27667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-fun 5448  df-fn 5449
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