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Theorem fn0 5210
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 5189 . . 3  |-  ( F  Fn  (/)  ->  Rel  F )
2 fndm 5190 . . 3  |-  ( F  Fn  (/)  ->  dom  F  =  (/) )
3 reldm0 4725 . . . 4  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
43biimpar 293 . . 3  |-  ( ( Rel  F  /\  dom  F  =  (/) )  ->  F  =  (/) )
51, 2, 4syl2anc 406 . 2  |-  ( F  Fn  (/)  ->  F  =  (/) )
6 fun0 5149 . . . 4  |-  Fun  (/)
7 dm0 4721 . . . 4  |-  dom  (/)  =  (/)
8 df-fn 5094 . . . 4  |-  ( (/)  Fn  (/) 
<->  ( Fun  (/)  /\  dom  (/)  =  (/) ) )
96, 7, 8mpbir2an 909 . . 3  |-  (/)  Fn  (/)
10 fneq1 5179 . . 3  |-  ( F  =  (/)  ->  ( F  Fn  (/)  <->  (/)  Fn  (/) ) )
119, 10mpbiri 167 . 2  |-  ( F  =  (/)  ->  F  Fn  (/) )
125, 11impbii 125 1  |-  ( F  Fn  (/)  <->  F  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1314   (/)c0 3331   dom cdm 4507   Rel wrel 4512   Fun wfun 5085    Fn wfn 5086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-fun 5093  df-fn 5094
This theorem is referenced by:  mpt0  5218  f0  5281  f00  5282  f0bi  5283  f1o00  5368  fo00  5369  tpos0  6137  ixp0x  6586  0fz1  9776
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