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Theorem fn0 5656
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 5635 . . 3  |-  ( F  Fn  (/)  ->  Rel  F )
2 fndm 5636 . . 3  |-  ( F  Fn  (/)  ->  dom  F  =  (/) )
3 reldm0 5014 . . . 4  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
43biimpar 487 . . 3  |-  ( ( Rel  F  /\  dom  F  =  (/) )  ->  F  =  (/) )
51, 2, 4syl2anc 665 . 2  |-  ( F  Fn  (/)  ->  F  =  (/) )
6 fun0 5601 . . . 4  |-  Fun  (/)
7 dm0 5010 . . . 4  |-  dom  (/)  =  (/)
8 df-fn 5547 . . . 4  |-  ( (/)  Fn  (/) 
<->  ( Fun  (/)  /\  dom  (/)  =  (/) ) )
96, 7, 8mpbir2an 928 . . 3  |-  (/)  Fn  (/)
10 fneq1 5625 . . 3  |-  ( F  =  (/)  ->  ( F  Fn  (/)  <->  (/)  Fn  (/) ) )
119, 10mpbiri 236 . 2  |-  ( F  =  (/)  ->  F  Fn  (/) )
125, 11impbii 190 1  |-  ( F  Fn  (/)  <->  F  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437   (/)c0 3704   dom cdm 4796   Rel wrel 4801   Fun wfun 5538    Fn wfn 5539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-fun 5546  df-fn 5547
This theorem is referenced by:  mpt0  5666  f0  5724  f00  5725  f0bi  5726  f1o00  5807  fo00  5808  fconstfvOLD  6086  tpos0  6958  ixp0x  7505  0fz1  11770  hashf1  12568  fuchom  15809  grpinvfvi  16650  mulgfval  16702  mulgfvi  16705  symgplusg  16973  0frgp  17372  invrfval  17844  psrvscafval  18557  tmdgsum  21052  deg1fvi  22976  hon0  27388  fnchoice  37266  dvnprodlem3  37706
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