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Theorem fn0 5335
Description: A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fn0  |-  ( F  Fn  (/)  <->  F  =  (/) )

Proof of Theorem fn0
StepHypRef Expression
1 fnrel 5314 . . 3  |-  ( F  Fn  (/)  ->  Rel  F )
2 fndm 5315 . . 3  |-  ( F  Fn  (/)  ->  dom  F  =  (/) )
3 reldm0 4845 . . . 4  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
43biimpar 297 . . 3  |-  ( ( Rel  F  /\  dom  F  =  (/) )  ->  F  =  (/) )
51, 2, 4syl2anc 411 . 2  |-  ( F  Fn  (/)  ->  F  =  (/) )
6 fun0 5274 . . . 4  |-  Fun  (/)
7 dm0 4841 . . . 4  |-  dom  (/)  =  (/)
8 df-fn 5219 . . . 4  |-  ( (/)  Fn  (/) 
<->  ( Fun  (/)  /\  dom  (/)  =  (/) ) )
96, 7, 8mpbir2an 942 . . 3  |-  (/)  Fn  (/)
10 fneq1 5304 . . 3  |-  ( F  =  (/)  ->  ( F  Fn  (/)  <->  (/)  Fn  (/) ) )
119, 10mpbiri 168 . 2  |-  ( F  =  (/)  ->  F  Fn  (/) )
125, 11impbii 126 1  |-  ( F  Fn  (/)  <->  F  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353   (/)c0 3422   dom cdm 4626   Rel wrel 4631   Fun wfun 5210    Fn wfn 5211
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-fun 5218  df-fn 5219
This theorem is referenced by:  mpt0  5343  f0  5406  f00  5407  f0bi  5408  f1o00  5496  fo00  5497  tpos0  6274  ixp0x  6725  0fz1  10044
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