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Theorem f00 5581
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 5549 . . . . 5  |-  ( F : A --> (/)  ->  Fun  F )
2 frn 5553 . . . . . . 7  |-  ( F : A --> (/)  ->  ran  F 
C_  (/) )
3 ss0 3656 . . . . . . 7  |-  ( ran 
F  C_  (/)  ->  ran  F  =  (/) )
42, 3syl 16 . . . . . 6  |-  ( F : A --> (/)  ->  ran  F  =  (/) )
5 dm0rn0 5043 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
64, 5sylibr 212 . . . . 5  |-  ( F : A --> (/)  ->  dom  F  =  (/) )
7 df-fn 5409 . . . . 5  |-  ( F  Fn  (/)  <->  ( Fun  F  /\  dom  F  =  (/) ) )
81, 6, 7sylanbrc 657 . . . 4  |-  ( F : A --> (/)  ->  F  Fn  (/) )
9 fn0 5518 . . . 4  |-  ( F  Fn  (/)  <->  F  =  (/) )
108, 9sylib 196 . . 3  |-  ( F : A --> (/)  ->  F  =  (/) )
11 fdm 5551 . . . 4  |-  ( F : A --> (/)  ->  dom  F  =  A )
1211, 6eqtr3d 2467 . . 3  |-  ( F : A --> (/)  ->  A  =  (/) )
1310, 12jca 529 . 2  |-  ( F : A --> (/)  ->  ( F  =  (/)  /\  A  =  (/) ) )
14 f0 5580 . . 3  |-  (/) : (/) --> (/)
15 feq1 5530 . . . 4  |-  ( F  =  (/)  ->  ( F : A --> (/)  <->  (/) : A --> (/) ) )
16 feq2 5531 . . . 4  |-  ( A  =  (/)  ->  ( (/) : A --> (/)  <->  (/) : (/) --> (/) ) )
1715, 16sylan9bb 692 . . 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 1362    C_ wss 3316   (/)c0 3625   dom cdm 4827   ran crn 4828   Fun wfun 5400    Fn wfn 5401   -->wf 5402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-fun 5408  df-fn 5409  df-f 5410
This theorem is referenced by:  cantnff  7870  0wrd0  12236  supcvg  13300  ram0  14065  itgsubstlem  21361  uhgra0v  23066  usgra0v  23112  usgra1v  23130  ismgm  23629  wlkv0  30134
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