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Theorem fnimaeq0 5244
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 4901 . 2  |-  ( ( F " B )  =  (/)  <->  ( dom  F  i^i  B )  =  (/) )
2 incom 3268 . . . 4  |-  ( dom 
F  i^i  B )  =  ( B  i^i  dom 
F )
3 fndm 5222 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
43sseq2d 3127 . . . . . 6  |-  ( F  Fn  A  ->  ( B  C_  dom  F  <->  B  C_  A
) )
54biimpar 295 . . . . 5  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  B  C_  dom  F )
6 df-ss 3084 . . . . 5  |-  ( B 
C_  dom  F  <->  ( B  i^i  dom  F )  =  B )
75, 6sylib 121 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( B  i^i  dom  F )  =  B )
82, 7syl5eq 2184 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( dom  F  i^i  B )  =  B )
98eqeq1d 2148 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( dom  F  i^i  B )  =  (/)  <->  B  =  (/) ) )
101, 9syl5bb 191 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( ( F " B )  =  (/)  <->  B  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    i^i cin 3070    C_ wss 3071   (/)c0 3363   dom cdm 4539   "cima 4542    Fn wfn 5118
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fn 5126
This theorem is referenced by: (None)
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