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Theorem f00 5765
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 5731 . . . . 5  |-  ( F : A --> (/)  ->  Fun  F )
2 frn 5735 . . . . . . 7  |-  ( F : A --> (/)  ->  ran  F 
C_  (/) )
3 ss0 3816 . . . . . . 7  |-  ( ran 
F  C_  (/)  ->  ran  F  =  (/) )
42, 3syl 16 . . . . . 6  |-  ( F : A --> (/)  ->  ran  F  =  (/) )
5 dm0rn0 5217 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
64, 5sylibr 212 . . . . 5  |-  ( F : A --> (/)  ->  dom  F  =  (/) )
7 df-fn 5589 . . . . 5  |-  ( F  Fn  (/)  <->  ( Fun  F  /\  dom  F  =  (/) ) )
81, 6, 7sylanbrc 664 . . . 4  |-  ( F : A --> (/)  ->  F  Fn  (/) )
9 fn0 5698 . . . 4  |-  ( F  Fn  (/)  <->  F  =  (/) )
108, 9sylib 196 . . 3  |-  ( F : A --> (/)  ->  F  =  (/) )
11 fdm 5733 . . . 4  |-  ( F : A --> (/)  ->  dom  F  =  A )
1211, 6eqtr3d 2510 . . 3  |-  ( F : A --> (/)  ->  A  =  (/) )
1310, 12jca 532 . 2  |-  ( F : A --> (/)  ->  ( F  =  (/)  /\  A  =  (/) ) )
14 f0 5764 . . 3  |-  (/) : (/) --> (/)
15 feq1 5711 . . . 4  |-  ( F  =  (/)  ->  ( F : A --> (/)  <->  (/) : A --> (/) ) )
16 feq2 5712 . . . 4  |-  ( A  =  (/)  ->  ( (/) : A --> (/)  <->  (/) : (/) --> (/) ) )
1715, 16sylan9bb 699 . . 3  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( F : A --> (/)  <->  (/) : (/) --> (/) ) )
1814, 17mpbiri 233 . 2  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  F : A --> (/) )
1913, 18impbii 188 1  |-  ( F : A --> (/)  <->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    C_ wss 3476   (/)c0 3785   dom cdm 4999   ran crn 5000   Fun wfun 5580    Fn wfn 5581   -->wf 5582
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 5588  df-fn 5589  df-f 5590
This theorem is referenced by:  cantnff  8089  0wrd0  12528  supcvg  13626  ram0  14395  itgsubstlem  22184  uhgra0v  23986  usgra0v  24047  usgra1v  24066  wlkv0  24436  ismgm  24998  uhg0v  31846
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