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Theorem f00 5782
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 5748 . . . . 5  |-  ( F : A --> (/)  ->  Fun  F )
2 frn 5752 . . . . . . 7  |-  ( F : A --> (/)  ->  ran  F 
C_  (/) )
3 ss0 3795 . . . . . . 7  |-  ( ran 
F  C_  (/)  ->  ran  F  =  (/) )
42, 3syl 17 . . . . . 6  |-  ( F : A --> (/)  ->  ran  F  =  (/) )
5 dm0rn0 5070 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
64, 5sylibr 215 . . . . 5  |-  ( F : A --> (/)  ->  dom  F  =  (/) )
7 df-fn 5604 . . . . 5  |-  ( F  Fn  (/)  <->  ( Fun  F  /\  dom  F  =  (/) ) )
81, 6, 7sylanbrc 668 . . . 4  |-  ( F : A --> (/)  ->  F  Fn  (/) )
9 fn0 5713 . . . 4  |-  ( F  Fn  (/)  <->  F  =  (/) )
108, 9sylib 199 . . 3  |-  ( F : A --> (/)  ->  F  =  (/) )
11 fdm 5750 . . . 4  |-  ( F : A --> (/)  ->  dom  F  =  A )
1211, 6eqtr3d 2465 . . 3  |-  ( F : A --> (/)  ->  A  =  (/) )
1310, 12jca 534 . 2  |-  ( F : A --> (/)  ->  ( F  =  (/)  /\  A  =  (/) ) )
14 f0 5781 . . 3  |-  (/) : (/) --> (/)
15 feq1 5728 . . . 4  |-  ( F  =  (/)  ->  ( F : A --> (/)  <->  (/) : A --> (/) ) )
16 feq2 5729 . . . 4  |-  ( A  =  (/)  ->  ( (/) : A --> (/)  <->  (/) : (/) --> (/) ) )
1715, 16sylan9bb 704 . . 3  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( F : A --> (/)  <->  (/) : (/) --> (/) ) )
1814, 17mpbiri 236 . 2  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  F : A --> (/) )
1913, 18impbii 190 1  |-  ( F : A --> (/)  <->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    C_ wss 3436   (/)c0 3761   dom cdm 4853   ran crn 4854   Fun wfun 5595    Fn wfn 5596   -->wf 5597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-fun 5603  df-fn 5604  df-f 5605
This theorem is referenced by:  cantnff  8187  0wrd0  12697  supcvg  13913  ram0  14979  itgsubstlem  22998  uhgra0v  25035  usgra0v  25096  usgra1v  25115  wlkv0  25486  ismgmOLD  26046  uhgr0v  38994  usgr1vr  39105  uhg0v  39308
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