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Theorem f0 5406
Description: The empty function. (Contributed by NM, 14-Aug-1999.)
Assertion
Ref Expression
f0  |-  (/) : (/) --> A

Proof of Theorem f0
StepHypRef Expression
1 eqid 2177 . . 3  |-  (/)  =  (/)
2 fn0 5335 . . 3  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
31, 2mpbir 146 . 2  |-  (/)  Fn  (/)
4 rn0 4883 . . 3  |-  ran  (/)  =  (/)
5 0ss 3461 . . 3  |-  (/)  C_  A
64, 5eqsstri 3187 . 2  |-  ran  (/)  C_  A
7 df-f 5220 . 2  |-  ( (/) :
(/) --> A  <->  ( (/)  Fn  (/)  /\  ran  (/)  C_  A ) )
83, 6, 7mpbir2an 942 1  |-  (/) : (/) --> A
Colors of variables: wff set class
Syntax hints:    = wceq 1353    C_ wss 3129   (/)c0 3422   ran crn 4627    Fn wfn 5211   -->wf 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-fn 5219  df-f 5220
This theorem is referenced by:  f00  5407  f0bi  5408  f10  5495  map0g  6687  ac6sfi  6897  0met  13820
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