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Theorem f0dom0 5784
Description: A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
Assertion
Ref Expression
f0dom0  |-  ( F : X --> Y  -> 
( X  =  (/)  <->  F  =  (/) ) )

Proof of Theorem f0dom0
StepHypRef Expression
1 feq2 5729 . . . 4  |-  ( X  =  (/)  ->  ( F : X --> Y  <->  F : (/) --> Y ) )
2 f0bi 5783 . . . . 5  |-  ( F : (/) --> Y  <->  F  =  (/) )
32biimpi 197 . . . 4  |-  ( F : (/) --> Y  ->  F  =  (/) )
41, 3syl6bi 231 . . 3  |-  ( X  =  (/)  ->  ( F : X --> Y  ->  F  =  (/) ) )
54com12 32 . 2  |-  ( F : X --> Y  -> 
( X  =  (/)  ->  F  =  (/) ) )
6 feq1 5728 . . . 4  |-  ( F  =  (/)  ->  ( F : X --> Y  <->  (/) : X --> Y ) )
7 dm0 5068 . . . . 5  |-  dom  (/)  =  (/)
8 fdm 5750 . . . . 5  |-  ( (/) : X --> Y  ->  dom  (/)  =  X )
97, 8syl5reqr 2485 . . . 4  |-  ( (/) : X --> Y  ->  X  =  (/) )
106, 9syl6bi 231 . . 3  |-  ( F  =  (/)  ->  ( F : X --> Y  ->  X  =  (/) ) )
1110com12 32 . 2  |-  ( F : X --> Y  -> 
( F  =  (/)  ->  X  =  (/) ) )
125, 11impbid 193 1  |-  ( F : X --> Y  -> 
( X  =  (/)  <->  F  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437   (/)c0 3767   dom cdm 4854   -->wf 5597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-fun 5603  df-fn 5604  df-f 5605
This theorem is referenced by:  swrdn0  12771  elfrlmbasn0  19256  mavmulsolcl  19507
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