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Theorem f00 5309
Description: A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
Assertion
Ref Expression
f00  |-  ( F : A --> (/)  <->  ( F  =  (/)  /\  A  =  (/) ) )

Proof of Theorem f00
StepHypRef Expression
1 ffun 5270 . . . . 5  |-  ( F : A --> (/)  ->  Fun  F )
2 frn 5276 . . . . . . 7  |-  ( F : A --> (/)  ->  ran  F 
C_  (/) )
3 ss0 3398 . . . . . . 7  |-  ( ran 
F  C_  (/)  ->  ran  F  =  (/) )
42, 3syl 14 . . . . . 6  |-  ( F : A --> (/)  ->  ran  F  =  (/) )
5 dm0rn0 4751 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
64, 5sylibr 133 . . . . 5  |-  ( F : A --> (/)  ->  dom  F  =  (/) )
7 df-fn 5121 . . . . 5  |-  ( F  Fn  (/)  <->  ( Fun  F  /\  dom  F  =  (/) ) )
81, 6, 7sylanbrc 413 . . . 4  |-  ( F : A --> (/)  ->  F  Fn  (/) )
9 fn0 5237 . . . 4  |-  ( F  Fn  (/)  <->  F  =  (/) )
108, 9sylib 121 . . 3  |-  ( F : A --> (/)  ->  F  =  (/) )
11 fdm 5273 . . . 4  |-  ( F : A --> (/)  ->  dom  F  =  A )
1211, 6eqtr3d 2172 . . 3  |-  ( F : A --> (/)  ->  A  =  (/) )
1310, 12jca 304 . 2  |-  ( F : A --> (/)  ->  ( F  =  (/)  /\  A  =  (/) ) )
14 f0 5308 . . 3  |-  (/) : (/) --> (/)
15 feq1 5250 . . . 4  |-  ( F  =  (/)  ->  ( F : A --> (/)  <->  (/) : A --> (/) ) )
16 feq2 5251 . . . 4  |-  ( A  =  (/)  ->  ( (/) : A --> (/)  <->  (/) : (/) --> (/) ) )
1715, 16sylan9bb 457 . . 3  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( F : A --> (/)  <->  (/) : (/) --> (/) ) )
1814, 17mpbiri 167 . 2  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  F : A --> (/) )
1913, 18impbii 125 1  |-  ( F : A --> (/)  <->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1331    C_ wss 3066   (/)c0 3358   dom cdm 4534   ran crn 4535   Fun wfun 5112    Fn wfn 5113   -->wf 5114
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122
This theorem is referenced by:  dom0  6725
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