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Theorem fn0 5530
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 5509 . . 3  |-  ( F  Fn  (/)  ->  Rel  F )
2 fndm 5510 . . 3  |-  ( F  Fn  (/)  ->  dom  F  =  (/) )
3 reldm0 5057 . . . 4  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
43biimpar 485 . . 3  |-  ( ( Rel  F  /\  dom  F  =  (/) )  ->  F  =  (/) )
51, 2, 4syl2anc 661 . 2  |-  ( F  Fn  (/)  ->  F  =  (/) )
6 fun0 5475 . . . 4  |-  Fun  (/)
7 dm0 5053 . . . 4  |-  dom  (/)  =  (/)
8 df-fn 5421 . . . 4  |-  ( (/)  Fn  (/) 
<->  ( Fun  (/)  /\  dom  (/)  =  (/) ) )
96, 7, 8mpbir2an 911 . . 3  |-  (/)  Fn  (/)
10 fneq1 5499 . . 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 1369   (/)c0 3637   dom cdm 4840   Rel wrel 4845   Fun wfun 5412    Fn wfn 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-fun 5420  df-fn 5421
This theorem is referenced by:  mpt0  5538  f0  5592  f00  5593  f0bi  5594  f1o00  5673  fo00  5674  fconstfv  5940  tpos0  6775  map0e  7250  ixp0x  7291  0fz1  11469  hashf1  12210  fuchom  14871  grpinvfvi  15579  mulgfval  15628  mulgfvi  15631  symgplusg  15894  0frgp  16276  invrfval  16765  psrvscafval  17461  tmdgsum  19666  deg1fvi  21556  hon0  25197  fnchoice  29751
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