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Theorem fnimaeq0 5020
Description: Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Assertion
Ref Expression
fnimaeq0  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( F " B )  =  (/)  <->  B  =  (/) ) )

Proof of Theorem fnimaeq0
StepHypRef Expression
1 imadisj 4687 . 2  |-  ( ( F " B )  =  (/)  <->  ( dom  F  i^i  B )  =  (/) )
2 incom 3129 . . . 4  |-  ( dom 
F  i^i  B )  =  ( B  i^i  dom 
F )
3 fndm 4998 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
43sseq2d 2973 . . . . . 6  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
54biimpar 281 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
6 df-ss 2931 . . . . 5  |-  ( B 
C_  dom  F  <->  ( B  i^i  dom  F )  =  B )
75, 6sylib 127 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( B  i^i  dom  F )  =  B )
82, 7syl5eq 2084 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( dom  F  i^i  B )  =  B )
98eqeq1d 2048 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( dom  F  i^i  B )  =  (/)  <->  B  =  (/) ) )
101, 9syl5bb 181 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( F " B )  =  (/)  <->  B  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    i^i cin 2916    C_ wss 2917   (/)c0 3224   dom cdm 4345   "cima 4348    Fn wfn 4897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fn 4905
This theorem is referenced by: (None)
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