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Theorem f00 5696
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 5664 . . . . 5  |-  ( F : A --> (/)  ->  Fun  F )
2 frn 5668 . . . . . . 7  |-  ( F : A --> (/)  ->  ran  F 
C_  (/) )
3 ss0 3771 . . . . . . 7  |-  ( ran 
F  C_  (/)  ->  ran  F  =  (/) )
42, 3syl 16 . . . . . 6  |-  ( F : A --> (/)  ->  ran  F  =  (/) )
5 dm0rn0 5159 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
64, 5sylibr 212 . . . . 5  |-  ( F : A --> (/)  ->  dom  F  =  (/) )
7 df-fn 5524 . . . . 5  |-  ( F  Fn  (/)  <->  ( Fun  F  /\  dom  F  =  (/) ) )
81, 6, 7sylanbrc 664 . . . 4  |-  ( F : A --> (/)  ->  F  Fn  (/) )
9 fn0 5633 . . . 4  |-  ( F  Fn  (/)  <->  F  =  (/) )
108, 9sylib 196 . . 3  |-  ( F : A --> (/)  ->  F  =  (/) )
11 fdm 5666 . . . 4  |-  ( F : A --> (/)  ->  dom  F  =  A )
1211, 6eqtr3d 2495 . . 3  |-  ( F : A --> (/)  ->  A  =  (/) )
1310, 12jca 532 . 2  |-  ( F : A --> (/)  ->  ( F  =  (/)  /\  A  =  (/) ) )
14 f0 5695 . . 3  |-  (/) : (/) --> (/)
15 feq1 5645 . . . 4  |-  ( F  =  (/)  ->  ( F : A --> (/)  <->  (/) : A --> (/) ) )
16 feq2 5646 . . . 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 1370    C_ wss 3431   (/)c0 3740   dom cdm 4943   ran crn 4944   Fun wfun 5515    Fn wfn 5516   -->wf 5517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-fun 5523  df-fn 5524  df-f 5525
This theorem is referenced by:  cantnff  7988  0wrd0  12366  supcvg  13431  ram0  14196  itgsubstlem  21648  uhgra0v  23391  usgra0v  23437  usgra1v  23455  ismgm  23954  wlkv0  30434
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