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Theorem f0dom0 5286
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 5226 . . . 4  |-  ( X  =  (/)  ->  ( F : X --> Y  <->  F : (/) --> Y ) )
2 f0bi 5285 . . . . 5  |-  ( F : (/) --> Y  <->  F  =  (/) )
32biimpi 119 . . . 4  |-  ( F : (/) --> Y  ->  F  =  (/) )
41, 3syl6bi 162 . . 3  |-  ( X  =  (/)  ->  ( F : X --> Y  ->  F  =  (/) ) )
54com12 30 . 2  |-  ( F : X --> Y  -> 
( X  =  (/)  ->  F  =  (/) ) )
6 feq1 5225 . . . 4  |-  ( F  =  (/)  ->  ( F : X --> Y  <->  (/) : X --> Y ) )
7 dm0 4723 . . . . 5  |-  dom  (/)  =  (/)
8 fdm 5248 . . . . 5  |-  ( (/) : X --> Y  ->  dom  (/)  =  X )
97, 8syl5reqr 2165 . . . 4  |-  ( (/) : X --> Y  ->  X  =  (/) )
106, 9syl6bi 162 . . 3  |-  ( F  =  (/)  ->  ( F : X --> Y  ->  X  =  (/) ) )
1110com12 30 . 2  |-  ( F : X --> Y  -> 
( F  =  (/)  ->  X  =  (/) ) )
125, 11impbid 128 1  |-  ( F : X --> Y  -> 
( X  =  (/)  <->  F  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316   (/)c0 3333   dom cdm 4509   -->wf 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097
This theorem is referenced by: (None)
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