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Theorem f0dom0 5204
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 5146 . . . 4  |-  ( X  =  (/)  ->  ( F : X --> Y  <->  F : (/) --> Y ) )
2 f0bi 5203 . . . . 5  |-  ( F : (/) --> Y  <->  F  =  (/) )
32biimpi 118 . . . 4  |-  ( F : (/) --> Y  ->  F  =  (/) )
41, 3syl6bi 161 . . 3  |-  ( X  =  (/)  ->  ( F : X --> Y  ->  F  =  (/) ) )
54com12 30 . 2  |-  ( F : X --> Y  -> 
( X  =  (/)  ->  F  =  (/) ) )
6 feq1 5145 . . . 4  |-  ( F  =  (/)  ->  ( F : X --> Y  <->  (/) : X --> Y ) )
7 dm0 4650 . . . . 5  |-  dom  (/)  =  (/)
8 fdm 5166 . . . . 5  |-  ( (/) : X --> Y  ->  dom  (/)  =  X )
97, 8syl5reqr 2135 . . . 4  |-  ( (/) : X --> Y  ->  X  =  (/) )
106, 9syl6bi 161 . . 3  |-  ( F  =  (/)  ->  ( F : X --> Y  ->  X  =  (/) ) )
1110com12 30 . 2  |-  ( F : X --> Y  -> 
( F  =  (/)  ->  X  =  (/) ) )
125, 11impbid 127 1  |-  ( F : X --> Y  -> 
( X  =  (/)  <->  F  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   (/)c0 3286   dom cdm 4438   -->wf 5011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-fun 5017  df-fn 5018  df-f 5019
This theorem is referenced by: (None)
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