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Theorem f0dom0 5769
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 5714 . . . 4  |-  ( X  =  (/)  ->  ( F : X --> Y  <->  F : (/) --> Y ) )
2 f0bi 5768 . . . . 5  |-  ( F : (/) --> Y  <->  F  =  (/) )
32biimpi 194 . . . 4  |-  ( F : (/) --> Y  ->  F  =  (/) )
41, 3syl6bi 228 . . 3  |-  ( X  =  (/)  ->  ( F : X --> Y  ->  F  =  (/) ) )
54com12 31 . 2  |-  ( F : X --> Y  -> 
( X  =  (/)  ->  F  =  (/) ) )
6 feq1 5713 . . . 4  |-  ( F  =  (/)  ->  ( F : X --> Y  <->  (/) : X --> Y ) )
7 dm0 5216 . . . . 5  |-  dom  (/)  =  (/)
8 fdm 5735 . . . . 5  |-  ( (/) : X --> Y  ->  dom  (/)  =  X )
97, 8syl5reqr 2523 . . . 4  |-  ( (/) : X --> Y  ->  X  =  (/) )
106, 9syl6bi 228 . . 3  |-  ( F  =  (/)  ->  ( F : X --> Y  ->  X  =  (/) ) )
1110com12 31 . 2  |-  ( F : X --> Y  -> 
( F  =  (/)  ->  X  =  (/) ) )
125, 11impbid 191 1  |-  ( F : X --> Y  -> 
( X  =  (/)  <->  F  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   (/)c0 3785   dom cdm 4999   -->wf 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5590  df-fn 5591  df-f 5592
This theorem is referenced by:  elfrlmbasn0  18591  mavmulsolcl  18848
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